Mode-locked pulsed laser system and method

ABSTRACT

Disclosed is a system and method for stabilizing the carrier-envelope phase of the pulses emitted by a femtosecond mode-locked laser by using the powerful tools of frequency-domain laser stabilization. Control of the pulse-to-pulse carrier-envelope phases was confirmed using temporal cross correlation. This phase stabilization locks the absolute frequencies emitted by the laser, which is used to perform absolute optical frequency measurements that were directly referenced to a stable microwave clock.

CROSS REFERENCE TO RELATED APPLICATIONS

[0001] This application claims the benefit of U.S. provisionalapplication serial number 60/193,287, filed Mar. 30, 2000, entitled“Direct Optical Frequency Synthesis and Phase Stabilization ofUltrashort Optical Pulses,” by Steven T. Cundiff, Scott A. Diddams, JohnL. Hall, and David J. Jones.

BACKGROUND

[0002] A. Field

[0003] The present disclosure pertains generally to lasers and moreparticularly to ultrafast mode-locked pulsed lasers. In one aspect thisdisclosure discusses carrier-envelope phase control of femtosecondmode-locked lasers and direct optical frequency synthesis

[0004] B. Background

[0005] 1. Introduction. Progress in femtosecond pulse generation hasmade it possible to generate optical pulses that are only a few cyclesin duration. [See G. Steinmeyer, D. H. Sutter, L. Gallmann, N.Matuschek, U. Keller, Science 286,1507 (1999); M. T. Asaki, C.-P. Huang,D. Garvey, J. Zhou, H. C. Kapteyn, M. M. Murnane, Opt. Lett. 18, 977(1993); U. Morgner, F. X. Kartner, S. H. Cho, Y. Chen, H. A. Haus, J. G.Fujimoto, E. P. Ippen, V. Scheuer, G. Angelow, T. Tschudi, Opt. Lett.24, 411 (1999); D. H. Sutter, G. Steinmeyer, L. Gallmann, N. Matuschek,F. Morier-Genoud, U. Keller, V. Scheuer, G. Angelow, T. Tschudi, Opt.Left. 24, 631 (1999)]. This has resulted in rapidly growing interest incontrolling the phase of the underlying carrier wave with respect to theenvelope. [See G. Steinmeyer, D. H. Sutter, L. Gallmann, N. Matuschek,U. Keller, Science 286,1507 (1999); L. Xu, Ch. Spielmann, A. Poppe, T.Brabec, F. Krausz, T. W. Hansch, Opt. Lett. 21, 2008 (1996); P.Dietrich, F. Krausz, P. B. Corkum, Opt. Lett. 25, 16 (2000); R. J.Jones, J.-C. Diels, J. Jasapara, W. Rudolph, Opt. Commun. 175,409(2000)]. The “absolute” carrier phase is normally not important inoptics; however, for such ultrashort pulses, it can have physicalconsequences. [See P. Dietrich, F. Krausz, P. B. Corkum, Opt. Lett. 25,16 (2000); C. G. Durfee, A. Rundquist, S. Backus, C. Heme, M. M.Murname, H. C. Kapteyn, Phys. Rev. Lett. 83, 2187 (1999)]. Concurrently,mode-locked lasers, which generate a train of ultrashort pulses, havebecome an important tool in precision optical frequency measurement.[See T. Udem, J. Reichert, R. Holzwarth, T. W. Hänsch, Phys. Rev. Lett.82, 3568 (1999); T. Udem, J. Reichert, R. Holzwarth, T. W. Hänsch, Opt.Lett. 24, 881 (1999); J. Reichert, R. Holzwarth, Th. Udem, T. W. Hänsch,Opt. Comm. 172, 59 (1999); S. A. Diddams, D. J. Jones, L.-S. Ma, S. T.Cundiff, J. L. Hall, Opt. Lett. 25, 186 (2000); S. A. Diddams, D. J.Jones, J. Ye, S. T. Cundiff, J. L. Hall, J. K. Ranka, R. S. Windeler, R.Holzwarth, T. Udem, T. W. Häsch, Phys. Rev. Lett. 84, 5102 (2000);Various schemes for using mode-locked lasers in optical frequencymetrology were recently discussed in H. R. Telle, G. Steinmeyer, A. E.Dunlop, J. Stenger, D. H. Sutter, U. Keller, Appl. Phys. B 69, 327(1999)]. There is a close connection between these two apparentlydisparate topics. This connection has been exploited in accordance withthe present invention to develop a frequency domain technique thatstabilizes the carrier phase with respect to the pulse envelope. Usingthe same technique, absolute optical frequency measurements wereperformed in accordance with the present invention using a singlemode-locked laser with the only input being a stable microwave clock.

[0006] Mode-locked lasers generate a repetitive train of ultrashortoptical pulses by fixing the relative phases of all of the lasinglongitudinal modes. [See A. E. Siegman, Lasers, (University ScienceBooks, Mill Valley Calif., 1986), p. 1041-1128]. Current mode-lockingtechniques are effective over such a large bandwidth that the resultingpulses can have a duration of 6 femtoseconds or shorter, i.e.,approximately two optical cycles. [See M. T. Asaki, C.-P. Huang, D.Garvey, J. Zhou, H. C. Kapteyn, M. M. Murnane, Opt. Lett. 18, 977(1993); U. Morgner, F. X. Kärtner, S. H. Cho, Y. Chen, H. A. Haus, J. G.Fujimoto, E. P. Ippen, V. Schenuer, G. Angelow, T. Tschudi, Opt. Lett.24, 411 (1999); D. H. Sutter, G. Steinmeyer, L. Gallmann, N. Matuschek,F. Morier-Genoud, U. Keller V. Scheuer, G. Angelow, T. Tschudi, Opt.Lett. 24, 631 (1999)]. With such ultrashort pulses, the relative phasebetween the peak of the pulse envelope and the underlying electric-fieldcarrier wave becomes relevant. In general, this phase is not constantfrom pulse-to-pulse because the group and phase velocities differ insidethe laser cavity (see FIG. 7A). To date, techniques of phase control offemtosecond pulses have employed time domain methods. [See L. Xu, Ch.Spielmann, A. Poppe, T. Brabec, F. Krausz, T. W. Hänsch, Opt. Lett. 21,2008 (1996)]. However, these techniques have not utilized activefeedback, and rapid dephasing occurs because of pulse energyfluctuations and other perturbations inside the cavity. Active controlof the relative carrier-envelope phase prepares a stable pulse-to-pulsephase relationship, as presented below, and will dramatically impactextreme nonlinear optics.

[0007] At the present, measurement of frequencies into the microwaveregime (tens of gigahertz) is straightforward thanks to the availabilityof high frequency counters and synthesizers. Historically, this has notalways been the case, with direct measurement being restricted to lowfrequencies. The current capability arose because an array of techniqueswas developed to make measurement of higher frequencies possible. [SeeG. E. Sterling and R. B. Monroe, The Radio Manual (Van Nostrand, NewYork, 1950)]. These techniques typically rely on heterodyning to producean easily measured frequency difference (zero-beating being the limit).The difficulty lay in producing an accurately known frequency to beat anunknown frequency against.

[0008] Measurement of optical frequencies (hundreds of terahertz) hasbeen in a similar primitive state until recently. This is because onlyfew “known” frequencies have been available and it has been difficult tobridge the gap between a known frequency and an arbitrary unknownfrequency of the gap exceeds tens of gigahertz (about 0.01% of theoptical frequency). Furthermore, establishing known optical frequencieswas itself difficult because an absolute measurement of frequency mustbe based on the time unit “second”, which is defined in terms of themicrowave frequency of a hyperfine transition of the cesium atom. Thisrequires a complex “clockwork” to connect optical frequencies to thosein the microwave region.

[0009] Optical frequencies have been used in measurement science sinceshortly after the invention of lasers. Comparison of a laser's frequencyof ˜5×10¹⁴ Hz with its ideal˜milliHertz linewidth, produced by thefundamental phase diffusion of spontaneous emission, reveals a potentialdynamic range of 10¹⁷ in resolution, offering one of the best tools fordiscovering new physics in “the next decimal place”. Nearly forty yearsof vigorous research in the many diverse aspects of this field by aworldwide community have resulted in exciting discoveries in fundamentalscience and development of enabling technologies. Some of the ambitiouslong-term goals in optical frequency metrology are just coming tofruition owing to a number of recent spectacular technological advances,most notably, the use of mode-locked lasers for optical frequencysynthesis. Other examples include laser frequency stabilization to oneHz and below [B. C. Young, F. C. Cruz, W. M. Itano, and J. C. Bergquist,Phys. Rev. Lett. 82, 3799-3802 (1999)], optical transitions observed ata few Hz linewidth (corresponding to a Q of 1.5×10¹⁴) [R. J. Rafac, B.C. Young, J. A. Beall, W. M. Itano, D. J. Wineland, and J. C. Bergquist,Phys. Rev. Lett. 85, 2462-2465 (2000)] and steadily improving accuracyof optical standards with a potential target of 10⁻¹⁸ for cold atom/ionsystems.

[0010] 2. Optical Frequency Synthesis and Metrology. Optical frequencymetrology broadly contributes to and profits from many areas in scienceand technology. At the core of this subject is the controlled and stablegeneration of coherent optical waves, i.e. optical frequency synthesis.This permits high precision and high resolution measurement of manyphysical quantities.

[0011] Below brief discussions are provided on these aspects of opticalfrequency metrology, with stable lasers and wide bandwidth opticalfrequency combs making up the two essential components in stablefrequency generation and measurement.

[0012] a. Establishment of standards. In 1967, just a few years afterthe invention of the laser, the international standard of time/frequencywas established, based on the F=4, m_(F)=0 F=3, m_(F)=0 transition inthe hyperfine structure of the ground state of ¹³³Cs. [See N. F. Ramsey,Journal of Res. of NBS 88, 301-320 (1983)]. The transition frequency isdefined to be 9,192,631,770 Hz. The resonance Q of ˜10⁸ is set by thelimited coherent interaction time between matter and field. Much efforthas been invested in extending the coherent atom-field interaction timeand in controlling the first and second order Doppler shifts. Recentadvances in the laser cooling and trapping technology have led to thepractical use of laser-slowed atoms, and a hundred-fold resolutionenhancement. With the reduced velocities, Doppler effects have also beengreatly reduced. Cs clocks based on atomic fountains are now operationalwith reported accuracy of 3×10⁻¹⁵ and short term stability of 1×10⁻¹³ at1 second, limited by the frequency noise of the local rf crystaloscillator. [See C. Santarelli, P. Laurent, P. Lemonde, A. Clairon, A.G. Mann, S. Chang, A. N. Luiten, and C. Salomon, Phys. Rev. Lett. 82,4619-4622 (1999)]. Through similar technologies, single ions,laser-cooled and trapped in an electromagnetic field, are now alsoexcellent candidates for radio frequency/microwave standards with ademonstrated frequency stability approaching 3×10⁻¹³ at 1 second. [SeeSullivan, D. B., J. C. Bergquist, J. J. Bollinger, R. E. Drullinger, W.M. Itano, S. R. Jefferts, W. D. Lee, D. Meekhof, T. E. Parker, F. L.Walls, D. J. Wineland, “Primary Atomic Frequency Standards at NIST”, J.Res. NIST, 2001, 106(1) pp47-63; D. J. Berkeland, J. D. Miller, J. C.Berquist, W. M. Itano, and D. J. Wineland, Phys. Rev. Lett. 80,2089-2092 (1998)]. More compact, less expensive, (and less accurate)atomic clocks use cesium or rubidium atoms in a glass cell, equippedwith all essential clock mechanisms, including optical pumping (atompreparation), microwave circuitry for exciting the clock transition, andoptical detection. The atomic hydrogen maser is another mature andpractical device that uses the radiation emitted by atoms directly. [SeeH. M. Goldenberg, D. Kleppner, and N. F. Ramsey, Phys. Rev. Lett. 8, 361(1960)]. Although it is less accurate than the cesium standard, ahydrogen maser can realize exceptional short-term stability.

[0013] The development of optical frequency standards has been anextremely active field since the invention of lasers, which provide thecoherent radiation necessary for precision spectroscopy. The coherentinteraction time, the determining factor of the spectral resolution inmany cases, is in fact comparable in both optical and rf domains. Theoptical part of the electromagnetic spectrum provides higher operatingfrequencies. Therefore the quality factor, Q, of an optical clocktransition is expected to be a few orders higher than that available inthe microwave domain. A superior Q factor helps to improve all threeessential characteristics of a frequency standard, namely accuracy,reproducibility and stability. Accuracy refers to the objective propertyof a standard to identify the frequency of a natural quantum transition,idealized to the case that the atoms or the molecules are at rest andfree of any perturbation. Reproducibility measures the repeatability ofa frequency standard for different realizations, signifying adequatemodeling of observed operating parameters and independence fromuncontrolled operating conditions. Stability indicates the degree towhich the frequency stays constant after operation has started. Ideally,a stabilized laser can achieve a fractional frequency stability${\frac{\delta \quad v}{v} = {\frac{1}{Q}\frac{1}{S/N}\frac{1}{\sqrt{\tau}}}},$

[0014] where S/N is the recovered signal-to-noise ratio of the resonanceinformation, and τ is the averaging time. Clearly it is desirable toenhance both the resolution and sensitivity of the detected resonance,as these control the time scale necessary for a given measurementprecision. The reward is enormous: enhancing the Q (or S/N) by a factorof ten reduces the averaging time by a factor of 100.

[0015] The nonlinear nature of a quantum absorption process, whilelimiting the attainable S/N, permits sub-Doppler resolution. Specialoptical techniques invented in the 70's and 80's for sub-Dopplerresolution include saturated absorption spectroscopy, two-photonspectroscopy, optical Ramsey fringes, optical double resonance, quantumbeats and laser cooling and trapping. Cold samples offer the truepossibility to observe the rest frame atomic frequency. Sensitivedetection techniques, such as polarization spectroscopy, electronshelving (quantum jump), and frequency modulation optical heterodynespectroscopy, were also invented during the same period, leading to anabsorption sensitivity of 1×10⁻⁸ and the ability to split a MHz scalelinewidth typically by a factor of 10⁴-10⁵, at an averaging time of ˜1s. All these technological advances paved the way for sub-Hertzstabilization of super-coherent optical local oscillators.

[0016] To effectively use a laser as a stable and accurate optical localoscillator, active frequency control is needed, owing to the strongcoupling between the laser frequency and the laser parameters. Thesimultaneous use of quantum absorbers and an optical cavity offers anattractive laser stabilization system. A passive reference cavity bringsthe benefit of a linear response allowing use of sufficient power toachieve a high S/N. On one hand, a laser pre-stabilized by a cavityoffers a long phase coherence time, reducing the need for frequentinterrogations of the quantum absorber. In other words, the laserlinewidth over a short time scale is narrower than the chose atomictransition width and thus the information of the natural resonance canbe recovered with an optimal S/N and the long averaging time translatesinto a finer examination of the true line center. On the other hand, thequantum absorber's resonance basically eliminates inevitable driftsassociated with material standards, such as a cavity. Frequencystability in the 10⁻¹⁶ domain has been measured with a cavity-stabilizedlaser. [See C. Salomon, D. Hils, and J. L. Hall, J. Opt. Soc. Am. B 5,1576-1587 (1988)]. The use of frequency modulation for cavity/laser lockhas become a standard laboratory practice. [See R. W. P. Drever, J. L.Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward,App. Phys. B 31, 97-105 (1983)]. Tunability of such a cavity/lasersystem can be obtained by techniques such as the frequency-offsetoptical phase-locked-loop (PLL).

[0017] A broad spectrum of lasers have been stabilized, from earlyexperiments with gas lasers (He—Ne, CO₂, Ar⁺, etc.) to more recenttunable dye lasers, optically pumper solid-state lasers Ti:Sapphire,YAG, etc.) and diode lasers. Usually one or several atomic or moleculartransitions are located within the tuning range of the laser to bestabilized. The use of molecular ro-vibrational lines for laserstabilization has been very successful in the infrared, using moleculessuch as CH₄, CO₂ and OsO₄. [See T. J. Quinn, Metrologia 36, 211-244(1999)]. Their natural linewidths range below a kilohertz, as limited bymolecular fluorescent decay. Useable linewidths are usually 10 kHz dueto transit of molecules through the light beam. Transitions to higherlevels of these fundamental ro-vibrational states, usually termedovertone bands, extend these ro-vibrational spectra well into thevisible with similar ˜kHz potential linewidths. Until recently, the richspectra of the molecular overtone bands have not been adopted assuitable frequency references in the visible due to their smalltransition strengths. [See M. Delabachelerie, K. Nakagawa, and M.Ohstru, Opt. Lett. 19, 840-842 (1994)]. However, with one of the mostsensitive absorption techniques, which combines frequency modulationwith cavity enhancement, an excellent S/N for these weak but narrowovertone lines can be achieved [J. Ye, L. S. Ma, and J. L. Hall, J. Opt.Soc. Am. B 15, 6-15 (1998)], enabling the use of molecular overtones asstandards in the visible. [See J. Ye, L. S. Ma, and J. L. Hall, IEEETrans. Instrum. Meas. 46, 178-182 (1997); J. Ye, L. S. Ma, and J. L.Hall, J. Opt. Soc. Am. B 17, 927-931 (2000)].

[0018] Systems based on cold absorber samples potentially offer thehighest quality optical frequency sources, mainly due to the drasticreductions of linewidth and velocity-related systematic errors. Forexample, a few Hz linewidth on the uv transition of Hg⁺ was recentlyobserved at NIST. [See R. J. Rafac, B. C. Young, J. A. Beall, W. M.Itano, D. J. Wineland, and J. C. Bergquist, Phys. Rev. Lett. 85,2462-2465 (2000)]. Current activity on single ion systems includes Sr⁺[J. E. Bernard, A. A. Madej, L. Marmet, B. G. Whitford, K. J. Siemsen,and S. Cundy, Phys. Rev. Lett. 82, 3228-3231 (1999)] Yb⁺ [M. Roberts, P.Taylor, G. P. Barwood, P. Gill, H. A. Klein, and W. R. C. Rowley, Phys.Rev. Lett. 78, 1876-1879 (1997)] and In⁺, [E. Peik, J. Abel, T. Becker,J. von Zanthier, and H. Walther, Phys. Rev. A 60, 439-449 (1999)]. Oneof the early NIST proposals of using atomic fountains for opticalfrequency standards [J. L. Hall, M. Zhu, and P. Buch, J. Opt. Soc. Am. B6, 2194-2205 (1989)] has resulted in investigation of the neutral atomsMg, Ca, Sr, Ba, and Ag. These systems could offer ultimate frequencystandards free from virtually all of the conventional shifts andbroadenings, to the level of one part in 10¹⁶-10¹⁸. Considerations of apractical system must always include its cost, size and degree ofcomplexity. Compact and low cost systems can be competitive even thoughtheir performance may be 10-fold worse compared with the ultimatesystem. One such system is Nd:YAG laser stabilized on HCCD at 1064 nm oron I₂ (after frequency doubling) at 532 nm, with a demonstratedstability level of 4×10⁻¹⁵ at 300-s averaging time. [See J. Ye, L.Robertsson, S. Picard, L. S. Ma, and J. L. Hall, IEEE Trans. Instrum.Meas. 48, 544-549 (1999); J. L. Hall, L. S. Ma, M. Taubman, B. Tiemann,F. L. Hong, O. Pfister, and J. Ye, IEEE Trans. Instrum. Meas. 48,583-586 (1999)]. FIG. 1 summarizes some of the optical frequencystandards 100 that are either established or under active development.Also indicated is the spectral width of currently available opticalfrequency combs 102 generated by mode-locked lasers.

[0019] Accurate knowledge of the center of the resonance is essentialfor establishing standards. Collisions, electromagnetic fringe fields,residual Doppler effects, probe field wave-front curvature, and probepower can all produce undesired center shifts and linewidth broadening.Other physical interactions, and even distortion in the modulationwaveform, can produce asymmetry in the recovered signal line shape. Forexample, in frequency modulation spectroscopy, residual amplitudemodulation introduces unwanted frequency shifts and instability andtherefore needs to be controlled. [See J. L. Hall, J. Ye, L.-S. Ma, K.Vogel, and T. Dinneen, in Laser Spectroscopy XIII, edited by Z.-J. Wang,Z.-M. Zhang and Y.-Z. Wang (World Scientific, Sinagpore, 1998), p.75-80]. These issues must be addressed carefully before one can becomfortable talking about accuracy. A more fundamental issue related totime dilation of the reference system (second order Doppler effect) canbe solved in a controlled fashion, one simply knows the sample velocityaccurately (for example, by velocity selective Raman process), or thevelocity is brought down to a negligible level using cooling andtrapping techniques.

[0020] b. Application of Standards. The technology of laser frequencystabilization has been refined and simplified over the years and hasbecome an indispensable research tool in many modern laboratoriesinvolving optics. Research on laser stabilization has been and still ispushing the limits of measurement science. Indeed, a number of currentlyactive research projects on fundamental physical principles greatlybenefit from stable optical sources and need continued progress on laserstabilization. They include: laser test of fundamental principles [D.Hils and J. L. Hall, Phys. Rev. Lett. 64, 1697-1700 (1990)],gravitational wave detection [P. Fritschel, G. Gonzalez, B. Lantz, P.Saha, and M. Zucker, Phys. Rev. Lett. 80, 3181-3184 (1998)], quantumdynamics [H. Mabuchi, J. Ye, and H. J. Kimble, Appl. Phys. B 68,1095-1108 (1999)], atomic and molecular structure, and many more. Recentexperiments with hydrogen atoms have led to the best reported value forthe Rydberg constant and 1S-Lamb shift. [See T. Udem, A. Huber, B.Gross, J. Reichert, M. Prevedelli, M. Weitz, and T. W. Hänsch, Phys.Rev. Lett. 79, 2646-2649 (1997); C. Schwob, L. Jozefowski, B.deBeauvoir, L. Hilico, F. Nez, L. Julien, F. Biraben, O. Acef, and A.Clairon, Phys. Rev. Lett. 82, 4960-4963 (1999)]. Fundamental physicalconstants such as the fine-structure constant, ratio of Planck'sconstant to electron mass, and the electron-to-proton mass ratio arealso being determined with increasing precision using improved precisionlaser tools. [See A. Peters, K. Y. Chung, B. Young. J. Hensley, and S.Chu, Philos. Trans. R. Soc. Lond. Ser. A 335, 2223-2233 (1997)]. Usingextremely stable phase-coherent optical sources, we are entering anexciting era when picometer resolution can be achieved over a millionkilometer distance in space. In time-keeping, an optical frequency clockis expected eventually to replace the current microwave atomic clocks.In length metrology, the realization of the basic unit, the “metre”,relies on stable optical frequencies. In communications, opticalfrequency metrology provides stable frequency/wavelength referencegrids. [See T. Ikegami, S. Sudo, and Y. Sakai, Frequency stabilizationof semiconductor laser diodes (Artech House, Norwood, 1995)].

[0021] A list of just a few examples of stabilized cw tunable lasersincludes milliHertz linewidth stabilization (relative to a cavity) fordiode-pumped solid state lasers, tens of milliHertz linewidth forTi:Sapphire lasers and sub-Hertz linewidths for diode and dye lasers.Tight phase locking between different laser systems can be achieved [J.Ye and J. L. Hall, Opt. Lett. 24, 1838-1840 (1999)] even for diodelasers that have fast frequency noise.

[0022] c. Challenge of Opitical Frequency Measurement & Synthesis.Advances in optical frequency standards have resulted in the developmentof absolute and precise frequency measurement capability in the visibleand near-infrared spectral regions. A frequency reference can beestablished only after it has been phase-coherently compared and linkedwith other standards. As mentioned above, until recently opticalfrequency metrology has been restricted to the limited set of “known”frequencies, due to the difficulty in bridging the gap betweenfrequencies and the difficulty in establishing the “known” frequenciesthemselves.

[0023] The traditional frequency measurement takes asynthesis-by-harmonics approach. Such a synthesis chain is a complexsystem, involving several stages of stabilized transfer lasers,high-accuracy frequency references (in both optical and rf ranges), andnonlinear mixing elements. Phase-coherent optical frequency synthesischains linked to the cesium primary standard include Cs—HeNe/CH₄ (3.9μm) [K. M. Evenson, J. S. Wells, F. R. Petersen, B. L. Danielson, and G.W. Day, Appl. Phys. Lett. 22, 192 (1973); C. O. Weiss, G. Kramer, B.Lipphardt, and E. Garcia, IEEE J. Quantum Electron. 24, 1970-1972(1988)] and Cs—CO₂/OsO₄ (10 μm). [See A. Clairon, B. Dahmani, A.Filimon, and J. Rutman, IEEE Trans. Instrum. Meas. 34, 265-268 (1985)].Extension to HeNe/I₂ (576 nm) [C. R. Pollock, D. A. Jennings, F. R.Petersen, J. S. Wells, R. E. Drullinger, E. C. Beaty, and K. M. Evenson,Opt. Lett. 8, 133-135 (1983)] and HeNe/I₂ (633 nm) [D. A. Jennings, C.R. Pollock, F. R. Petersen, R. E. Drullinger, K. M. Evenson, J. S.Wells, J. L. Hall, and H. P. Layer, Opt. Lett. 8, 136-138 (1983); O.Acef, J. J. Zondy, M. Abed, D. G. Rovera, A. H. Gerard, A. Clairon, P.Laurent, Y. Millerioux, and P. Juncar, Opt. Commun. 97, 29-34 (1993)]lasers made use of one of these reference lasers (or the CO₂/CO₂ system[K. M. Evenson, J. S. Wells, F. R. Petersen, B. L. Danielson, and G. W.Day, Appl. Phys. Lett. 22, 192 (1973)]) as an intermediate. The firstwell-stabilized laser to be measured by a Cs-based frequency chain wasthe HeNe/CH₄ system at 88 THz. [See K. M. Evenson, J. S. Wells, F. R.Petersen, B. L. Danielson, and G. W. Day, Appl. Phys. Lett. 22, 192(1973)]. With interferometric determination of the associated wavelength[R. L. Barger and J. L. Hall, Appl. Phys. Lett. 22, 196-199 (1973)] interms of the existing wavelength standard based on krypton discharge,the work let to a definitive value for the speed of light, soonconfirmed by other laboratories using many different approaches.Redefinition of the unit of length by adopting c=299,792,458 m/s becamepossible with the extension of the direct frequency measurements to 473THz (HeNe/I₂ 633 nm system) 10 years later by a NIST 10-person team,creating a direct connection between the time and length units. Morerecently, with improved optical frequency standards based on cold atoms(Ca) [H. Schmatz, B. Lipphardt, J. Helmcke, F. Richle, and G. Zinner,Phys. Rev. Lett. 76, 18-21 (1996)] and single trapped ions (Sr⁺) [J. E.Bernard, A. A. Madej, L. Marmet, B. G. Whitford, K. J. Siemsen, and S.Cundy, Phys. Rev. Lett. 82, 3228-3231 (1999)], these traditionalfrequency chains have demonstrated measurement uncertainties at the 100Hz level.

[0024] Understandably, these frequency chains are large scale researchefforts requiring resources that can be provided by only a few nationallaboratories. Furthermore, the frequency chain can only cover somediscrete frequency marks in the optical spectrum. Difference frequenciesof many THz could still remain between a target frequency and a knownreference. These three issues have represented major obstacles to makingoptical frequency metrology a general laboratory tool. Severalapproaches have been proposed and tested as simple, reliable solutionsfor bridging large optical frequency gaps. Some popular schemes include:frequency interval bisection [H R. Telle, D. Meschede, and T. W. Hansch,Opt. Lett. 15, 532-534 (1990)], optical-parametric oscillators (OPO) [N.C. Wong, Opt. Lett. 15, 1129-1131 (1990)], optical comb generators [M.Kourogi, K. Nakagawa, and M. Ohtsu, IEEE J. Quantum Electron. 29,2693-2701 (1993); L. R. Brothers, D. Lee, and N. C. Wong, Opt. Lett. 19,245-247 (1994)], sum-and-difference generation in the near infrared [D.Van Baak and L. Hollberg, Opt. Lett. 19, 1586-1588 (1994)], frequencydivision by three [O. Pfister, M. Murtz, J. S. Wells, L. Hollberg, andJ. T. Murray, Opt. Lett. 21, 1387-1389 (1996); P. T. Nee and N. C. Wong,Opt. Lett. 23, 46-48 (1998)] and four wave mixing in laser diodes. [SeeC. Koch and H. R. Telle, J. Opt. Soc. Am. B 13, 1666-1678 (1996)]. Allof these techniques rely on the principle of difference-frequencysynthesis, in contrast to the frequency harmonic generation methodnormally used in traditional frequency chains. In the next section webriefly summarize these techniques, their operating principles andapplications. Generation of wide bandwidth optical frequency combs hasprovided the most direct and simple approach among these techniques, andit is the basis for the present invention.

[0025] 3. Traditional Approaches to Optical Frequency Synthesis.Although the potential for using mode-locked lasers in optical frequencysynthesis was recognized early [J. N. Eckstein, A. I. Ferguson, and T.W. Hänsch, Phys. Rev. Lett. 40, 847-850 (1978)], they did not providethe properties necessary for fulfilling this potential until recently.Consequently, an enormous effort has been invested over the last 40years in “traditional” approaches, which typically involve phasecoherently linked single frequency lasers. Traditional approaches tooptical frequency measurement can be divided into two sub-categories,one is synthesis by harmonic generation, and the other isdifference-frequency synthesis. The former method has a long history ofsuccess, at the expense of massive resources and system complexity. Thelater approach has been the focus of recent research, leading to systemsthat are more flexible, adaptive and efficient and is the subject of thepresent invention which involves the use of a wide bandwidth opticalfrequency comb generator.

[0026] a. Phase Coherent Chains (traditional frequency harmonicgeneration). The traditional frequency measurement takes asynthesis-by-harmonic approach. Harmonics, i.e., integer multiples, of astandard frequency are generated with a nonlinear element and the outputsignal of a higher-frequency oscillator is phase coherently linked toone of the harmonics. Tracking and counting of the beat note, or the useof a phase-locked loop (PLL), preserves the phase coherence at eachstage. Such phase-coherent frequency multiplication process is continuedto higher and higher frequencies until the measurement target in theoptical spectrum is reached. In the frequency region of microwave tomid-infrared, a harmonic mixer can perform frequency multiplication andfrequency mixing/phase comparison all by itself. “Cat's whisker” W—Sipoint contact microwave diodes, metal-insulator-metal (MIM) diodes andSchottky diodes have been used extensively for this purpose. In thenear-infrared to the visible (<1.5 μm), the efficiency of MIM diodesdecreases rapidly. Optical nonlinear crystals are better for harmonicgeneration in these spectral regions. Fast photodiodes perform frequencymixing (non-harmonic) and phase comparison. Such a synthesis chain is acomplex system, involving several stages of stabilized transfer lasers,high-accuracy frequency references (in both optical and rf ranges), andnonlinear mixing elements. A important limitation is that eachoscillator stage employs different lasing transitions and differentlaser technologies, so that reliable and cost effective designs areelusive.

[0027] 1) Local Oscillators and Phase Locked Loops. The most importantissue in frequency synthesis is the stability and accuracy associatedwith such frequency transfer processes. Successful implementation of asynthesis chain requires a set of stable local oscillators at variousfrequency stages. Maintaining phase coherence across the vast frequencygaps covered by the frequency chain demands that phase errors at eachsynthesis stage be eliminated or controlled. A more stable localoscillator offers a longer phase coherence time, making frequency/phasecomparison more tractable and reducing phase errors accumulated beforethe servo can decisively express control. Owing to the intrinsicproperty of the harmonic synthesis process, there are two mechanisms forfrequency/phase noise to enter the loop and limit the ultimateperformance. The first is additive noise, where a noisy local oscillatorcompromises the information from a particular phase comparison step. Thesecond, and more fundamental one, is the phase noise associated with thefrequency (really phase) multiplication process: the phase angle noiseincreases as the multiplication factor, hence the phase noise spectraldensity of the output signal from a frequency multiplier increases asthe square of the multiplication factor and so becomes progressivelyworse as the frequency increases in each stage of the chain. Low phasenoise microwave and laser local oscillators are therefore important inall PLL frequency synthesis schemes.

[0028] The role of the local oscillator in each stage of the frequencysynthesis chain is to take up the phase information from the lowerfrequency regions and pass it on to the next level, with appropriatenoise filtering, and to reestablish a stable amplitude. The process offrequency/phase transfer typically involves phase-locked loops (PLLs).Sometimes, frequency comparison is carried out with a frequency countermeasuring the difference in cycle numbers between two periodic signals,within a predetermined time period. As an intrinsic time domain deviceused to measure zero-crossings, a frequency counter is sensitive tosignals—and noise—in a large bandwidth and so can easily accumulatecounting errors owing to an insufficient signal-to-noise ratio. Even fora PLL, the possibility of cycle slipping is a serious issue. With aspecified signal to noise ratio and control bandwidth, one can estimatethe average time between successive cycle slips and thus know theexpected frequency counting error. For example, a 100 kHz measurementbandwidth requires a signal-to-noise ratio of 11 dB to achieve afrequency error of 1 Hz (1 cycle slip per 1 s). [See J. L. Hall, M.Taubman, S. A. Diddams, B. Tiemann, J. Ye, L. S. Ma, D. J. Jones, and S.T. Cundiff, in Laser Spectroscopy XIII, edited by R. Blatt, J. Eschner,D. Leibfried and F. Schmidt-Kaler (World Scientific, Singapore, 1999),p. 51-60].

[0029] One function of PLLs is to regenerate a weak signal from a noisybackground, providing spectral filtering and amplitude stabilization.This function is described as a “tracking filter.” Within the correctionbandwidth, the tracking filter frequency output follows the perceived rfinput sinewave's frequency. A voltage-controlled-oscillator (VCO)provides the PLL's output constant amplitude, the variable outputfrequency is guided by the correction error generated from the phasecomparison with the weak signal input. A tracking filter, consisting ofa VCO under PLL control, is ordinarily essential for producing reliablefrequency counting, with the regenerated signal able to support theunambiguous zero-crossing measurement for a frequency counter.

[0030] 2) Measurements Made With Phase Coherent Chains. As described inthe previous section, only a few phase-coherent optical frequencysynthesis chains have ever been implemented. Typically, some importantinfrared standards, such as the 3.39 μm (HeNe/CH₄) system and the 10 μm(CO₂/OsO₄) system are connected to the Cs standard first. Onceestablished, these references are then used to measure higher opticalfrequencies.

[0031] One of the first frequency chains was developed at NBS,connecting the frequency of a methane-stabilized HeNe laser to the Csstandard. [See K. M. Evenson, J. S. Wells, F. R. Petersen, B. L.Danielson, and G. W. Day, Appl. Phys. Lett. 22, 192 (1973)]. The chainstarted with a Cs-referenced Klystron oscillator at 10.6 GHz, with its7th harmonic linked to a second Klystron oscillator at 74.2 GHz. A HCNlaser at 0.89 THz was linked to the 12th harmonic of the second Klystronfrequency. The 12th harmonic of the HCN laser was connected to a H₂Olaser, whose frequency was tripled to connect to a CO₂ laser at 32.13THz. A second CO₂ laser frequency, at 29.44 THz, was linked to thedifference between the 32.13 THz CO₂ laser and the third harmonic of theHCN laser. The third harmonic of this second CO₂ laser finally reachedthe HeNe/CH₄ frequency at 88.3762 THz. The measured value of HeNe/CH₄frequency was later used in another experiment to determine thefrequency of iodine-stabilized HeNe laser at 633 nm, bridging the gapbetween infrared and visible radiation. [See D. A. Jennings, C. R.Pollock, F. R. Peterson, R. E. Drullinger, K. M. Evenson, J. S. Wells,J. L. Hall, and H. P. Layer, Opt. Lett. 8, 136-138 (1983)].

[0032] The important 10 μm spectral region covered by CO₂ lasers hasbeen the focus of several different frequency chains. [See C. O. Weiss,G. Kramer, B. Lipphardt, and E. Garcia, IEEE J. Quantum Electron. 24,1970-1972 (1988); A. Clairon, B. Dahmani, A. Filimon, and J. Rutman,IEEE Trans. Instrum. Meas. 34, 265-268 (1985); B. G. Whitford, App.Phys. B 35, 119-122 (1984)]: It is worth noting that in the Whitfordchain [B. G. Whitford, App. Phys. B 35, 119-122 (1984)] a substantialnumber of difference frequencies (generated between various CO₂ lasers)were used to bridge the intermediate frequency gaps, although thegeneral principle of the chain itself is still based on harmonicsynthesis. CO₂ lasers provided the starting point of most subsequentfrequency chains that reached the visible frequency spectrum. [See C. R.Pollock, D. A. Jennings, F. R. Petersen, J. S. Wells, R. E. Drullinger,E. C. Beaty, and K. M. Evenson, Opt. Lett. 8, 133-135 (1983); O. Acef,J. J. Zondy, M. Abed, D. G. Rovera, A. H. Gerard, A. Clairon, P.Laurent, Y. Millerioux, and P. Juncar, Opt. Commun 97, 29-34 (1993). F.Nez, M. D. Plimmer, S. Bourzeix, I. Julien, F. Birabert, R. Felder, O.Acef, J. J. Zondy, P. Laurent, A. Clairon, M. Abed, Y. Millerioux, andP. Juncar, Phys. Rev. Lett. 69, 2326-2329 (1992)]. As noted above, thesefrequency chains and measurements have led to the accurate knowledge ofthe speed of light, allowing international redefinition of the “Metre”,and establishment of many absolute frequency/wavelength standardsthroughout the IR/visible spectrum. More recently, with improved opticalfrequency standards based on cold atoms (Ca) [H. Schnatz, B. Lipphardt,J. Helmcke, F. Riehle, and G. Zinner, Phys. Rev. Lett. 76, 18-21 (1996)]and single trapped ions (Sr⁺) [J. E. Bernard, A. A. Madej, L. Marmet, B.G. Whitford, K. J. Siemsen, and S. Cundy, Phys. Rev. Lett. 82, 3228-3231(1999)], these traditional frequency measurement techniques havedemonstrated measurement uncertainties at the 100 Hz level, by directlylinking the Cs standard to the visible radiation in a single frequencychain.

[0033] 3) Shortcomings of this Traditional Approach. It is obvious thatsuch harmonic synthesis systems require a significant investment ofhuman and other resources. The systems need constant maintenance and canbe afforded only by national laboratories. Perhaps the most unsatisfyingaspect of harmonic chains is that they cover only a few discretefrequency marks in the optical spectrum. Therefore the systems work oncoincidental overlaps in target frequencies and are difficult to adaptto different tasks. Another limitation is the rapid increase of phasenoise (as n²) with the harmonic synthesis order (n).

[0034] b. Difference Frequency Synthesis. The difference-frequencygeneration approach borrows many frequency measurement techniquesdeveloped for the harmonic synthesis chains. Perhaps the biggestadvantage of difference frequency synthesis over the traditionalharmonic generation is that the system can be more flexible and compact,and yet have access to more frequencies. Five recent approaches aredisclosed below, with the frequency interval bisection and the opticalcomb generator being the most significant breakthroughs. The commontheme of these techniques is the ability to subdivide a large opticalfrequency interval into smaller portions with a known relationship tothe original frequency gap. The small frequency difference is thenmeasured to yield the value of the original frequency gap.

[0035] 1) Frequency Interval Bisection. Bisection of frequency intervalsis one of the most important concepts in the difference frequencygeneration. Coherent bisection of optical frequency generates thearithmetic average of two laser frequencies f₁ and f₂ by phase lockingthe second harmonic of a third laser at frequency f₃ to the sumfrequency of f₁ and f₂. These frequency-interval bisection stages can becascaded to provide difference-frequency division by 2^(th). Thereforeany target frequency can potentially be reached with a sufficient numberof bisection stages. Currently the fastest commercial photodetectors canmeasure heterodyne beats of some tens of GHz. Thus, six to ten cascadedbisection stages are required to connect a few hundred THz widefrequency interval with a measurable microwave frequency. Therefore thecapability of measuring a large beat frequency between two opticalsignals becomes ever more important, considering the number of bisectionstages that can be saved with a direct measurement. A powerfulcombination is to have an optical comb generator capable of measuring afew THz optical frequency differences as the last stage of the intervalbisection chain. It is worth noting that in a difference frequencymeasurement it is typical for all participating lasers to have theirfrequencies in a nearby frequency interval, thus simplifying systemdesign. Many optical frequency measurement schemes have been proposed,and some realized, using interval bisection. The most notableachievement so far has been by Hänsch's group at the Max-PlanckInstitute for Quantum Optics (MPQ) in Garching, where the idea forbisection originated. They used a phase locked chain of five frequencybisection stages to bridge the gap between the hydrogen 1S-2S resonancefrequency and the 28th harmonic of the HeNe/CH₄ standard at 3.39 μm,leading to the improved measurement of the Rydberg constant and thehydrogen ground state Lamb shift. [See T. Udem, A. Huber, B. Gross, J.Reichert, M. Prevedelli, M. Weitz, and T. W. Hänsch, Phys. Rev. Lett.79, 2646-2649 (1997)]. The chain started with a interval divider betweena 486 nm laser (one fourth of the frequency of the hydrogen 1S-2Sresonance) and the HeNe/CH₄. The rest of the chain successively reducedthe gap between this midpoint near 848 nm and the 4th harmonic ofHeNe/CH₄, a convenient spectral region where similar diode laser systemscan be employed, even though slightly different wavelengths arerequired.

[0036] 2) Optical Parametric Oscillators. The use of optical parametricoscillators (OPOs) for frequency division relies on parametric downconversion to convert an input optical signal into two coherentsubharmonic outputs, the signal and idler. These outputs are tunable andtheir linewidths are replicas of the input pump except for the quantumnoise added during the down conversion process. The OPO outputfrequencies, or the original pump frequency, can be precisely determinedby phase locking the difference frequency between the signal and idlerto a known microwave or infrared frequency.

[0037] In Wong's original proposal, OPO divider stages configured inparallel or serial were shown to provide the needed multi-step frequencydivision. [See N. C. Wong, Opt. Lett. 15, 1129-1131 (1990)]. However, nosuch cascaded systems have been realized so far, owing in part to thedifficulty of finding suitable nonlinear crystals for the OPO operationto work in different spectral regions, especially in the infrared. Thereis progress on the OPO-based optical frequency measurement schemes, mostnotably optical frequency division by 2 and 3 [S. Slyusarev, T. Ikegami,and S. Ohshima, Opt. Lett. 24, 1856-1858 (1999); A. Douillet, J. J.Zondy, A. Yelisseyev, S. Lobanov, and L. Isaenko, IEEE Trans. Ultrason.Ferroelectr. Freq. Control 47, 1127-1133 (2000)] that allow rapidreduction of a large frequency gap. Along with threshold-free differencefrequency generations in nonlinear crystals (discussed next), the OPOsystem provides direct access to calibrated tunable frequency sources inthe IR region (20-200 THz).

[0038] 3) Nonlinear Crystal Optics. This same principle, i.e.,phase-locking between the difference frequency while holding the sumfrequency a constant, leads to frequency measurement in the nearinfrared using nonlinear crystals for the sum-and-difference frequencygeneration. The sum of two frequencies in the near infrared can bematched to a visible frequency standard while the difference matches toa stable reference in the mid infrared. Another important technique isoptical frequency division by 3. This larger frequency ratio couldsimplify optical frequency chains while providing a convenientconnection between visible lasers and infrared standards. An additionalstage of mixing is needed to ensure the precise division ratio. [See O.Pfister, M. Murtz, J. S. Wells, L. Hollber, and J. T. Murray, Opt. Lett.21, 1387-1389 (1996)].

[0039] 4) Four Wave Mixing In Laser Diodes. Another approach todifference frequency generation relies on four-wave mixing. The idea [C.Koch and H. R. Telle, J. Opt. Soc. Am. B 13, 1666-1678 (1996)] is to usea laser diode as both a light source and an efficient nonlinear receiverto allow a four-wave mixing process to generate phase-coherent bisectionof a frequency interval of a few THz. The setup involved two externalcavity diode lasers (ω_(LD1) and ω_(LD2)), separated by 1-2 THz, thatare optically injected into a third diode laser for frequency mixing.When the frequency of the third diode laser (ω_(LD3)) was tuned near theinterval center of ω_(LD1) and ω_(LD2), the injection locking mechanismbecame effective to lock ω_(LD3) on the four-wave mixing product,ω_(LD1)+ω_(LD2)−ω_(LD3), leading to the interval bisection condition:ω_(LD3)=(ω_(Ld1)+ω_(LD2))/2. The bandwidth of this process is limited byphase matching in the mixing diode, and was found to be only a few THz.[See C. Koch and H. R. Telle, J. Opt. Soc. Am. B 13, 1666-1678 (1996)].

[0040] 5) Optical Frequency Comb Generators. One of the most promisingdifference frequency synthesis techniques is the generation of multi-THzoptical combs by placing an rf electro-optic modulator (EOM) in alow-loss optical cavity. [See M. Kourogi, K. Nakagawa, and M. Ohtsu,IEEE J. Quantum Electron. 29, 2693-2701 (1993)]. The optical cavityenhances modulation efficiency by resonating with the carrier frequencyand all subsequently generated sidebands, leading to a spectral comb offrequency-calibrated lines spanning a few THz. The schematic of such anoptical frequency comb generation process is shown in FIGS. 2A and 2B.The single frequency cw laser beam 200, as shown in FIG. 2A, is lockedon one of the resonance modes of the EOM cavity, with thefree-spectral-range frequency of the loaded cavity being an integermultiple of the EOM modulation frequency. The optical cavity comprisesmirrors 208, 210. The optical cavity includes a resonant electro-opticmodulator 204 that is driven by a modulator 202 having a modulationfrequency f_(m). The cavity output 206 produces a comb spectrum 212shown in FIG. 2B with an intensity profile of exp{—|k|π/βF} [M. Kourogi,K. Nakagawa, and M. Ohtsu, IEEE J. Quantum Electron. 29, 2693-2701(1993)], where k is the order of generated sideband from the originalcarrier, β is the EOM phase modulation index, and F is the loaded cavityfinesse. The uniformity of the comb frequency spacing was carefullyverified. [See K. Imai, Y. Zhao, M. Kourogi, B. Widiyatmoko, and M.Ohtsu, Opt. Lett. 24, 214-216 (1999)]. These optical frequency combgenerators (OFCG) have produced spectra extending a few tens of THz [K.Imai, M. Kourogi, and M. Ohtsu, IEEE J. Quantum Electron. 34, 54-60(1998)] nearly 10% of the optical carrier frequency 214. A group atJILA, including the inventors, developed unique OFGCs, one withcapability of single comb line selection [J. Ye, L. S. Ma, T. Day, andJ. L. Hall, Opt. Lett. 22, 301-303 (1997)] and the other with efficiencyenhancement via an integrated OPO/EOM system. [See S. A. Diddams, L. S.Ma, J. Ye, and J. L. Hall, Opt. Lett. 24, 1747-1749 (1999)].

[0041] OFCGs had an immediate impact on the field of optical frequencymeasurement. Kourogi and coworkers [K. Nakagawa, M. deLabachelerie, Y.Awaji, and M. Kourogi, J. Opt. Soc. Am. B 13, 2708-2714 (1996)] producedan optical frequency map (accurate to 10⁻⁹) in the telecommunicationband near 1.5 μm, using an OFCG that produced a 2-THz wide comb in thatwavelength region, connecting various molecular overtone transitionbands of C₂H₂ and HCN. The absolute frequency of the Cs D₂ transition at852 nm was measured against the fourth harmonic of the HeNe/CH₄standard, with an OFCG bridging the remaining frequency gap of 1.78 THz.[See Udem, J. Reichert, T. W. Hansch, and M. Kourogi, Phys. Rev. A 62,031801-031801-031801-031804 (2000)]. The JILA group used a OFCG tomeasure the absolute optical frequency of the iodine stabilized Nd:YAGfrequency near 532 nm. (See J. L. Hall, L. S. Ma, M. Taubman, B.Tiemann, F. L. Hong, O. Pfister, and J. Ye, IEEE Trans. Instrum. Meas.48, 583-586 (1999)]. The level scheme for the measurement 310 is shownin FIG. 3. The sum frequency 300 of a Ti:Sapphire laser stabilized onthe Rb two photon transition at 778 nm 302 and the frequency doubledNd:YAG laser 304 was compared against the frequency-doubled output of adiode laser 306 near 632 nm. The 660 GHz frequency gap between the reddiode frequency doubled output 306 and the iodine-stabilized HeNe laser308 at 633 nm was measured using the OFCG. [See J. Ye, L. S. Ma, T. Day,and J. L. Hall, Opt. Lett. 22, 301-303 (1997)].

[0042] An OFCG was also used in the measurement of the absolutefrequency of a Ne transition (1S₅ 2P₈) at 636.6 nm, relative to theHeNe/I₂ standard at 632.99 nm. [See P. Dubé, personal communication1997; J. Ye, Ph.D Thesis, U. of Colorado (1997)]. The lower level of thetransition is a metastable state. Therefore, the resonance can only beobserved in a discharged neon cell. The resonance has a naturallinewidth of 7.8 MHz. It can be easily broadened (due to unresolvedmagnetic sublevels) and its center frequency shifted by an externalmagnetic field. This line is therefore not a high quality referencestandard. However, it does have the potential of becoming a low cost andcompact frequency reference that offers a frequency calibration on theorder of 100 kHz. A red diode laser probing and inexpensive neon lampform such a system.

[0043] The frequency gap between the HeNe/I₂ standard and the neontransition is about 468 GHz, which can easily be measured with an OFCG.The HeNe laser, which is the carrier of the comb, is locked to the ¹²⁷I₂R(127) 11-5 component a₁₃. The neon 1S₅ 2P₈ transition frequency wasdetermined to be 473,143,829.76 (0.10) MHz.

[0044] The results obtained using OFCGs made the advantages of largerbandwidth very clear. However the bandwidth achievable by a traditionalOFCG is limited by cavity dispersion and modulation efficiency. Toachieve even larger bandwidth, mode-locked lasers were introduced, thustriggering a true revolution in optical frequency measurement.Mode-locked lasers are employed in accordance with the presentinvention.

SUMMARY

[0045] The ability to generate wide-bandwidth optical frequency combshas recently provided truly revolutionary advances. This patentdescribes the implementation of a method and apparatus for stabilizingthe carrier phase with respect to the envelope of the pulses emitted bya mode-locked laser. The method and apparatus employ frequency domaintechniques which result in a series of regularly spaced frequencies thatform a “comb” spanning the optical spectrum. The optical frequency combis generated by a mode-locked laser. The comb spacing is such that anyoptical frequency can be easily measured by heterodyning that opticalfrequency with a nearby comb line. Furthermore, it is possible todirectly reference the comb spacing and position to the microwave cesiumtime standard, thereby determining the absolute optical frequencies ofall of the comb lines. This recent advance in the optical frequencymeasurement technology has facilitated the realization of the ultimategoal of a practical optical frequency synthesizer: it forms aphase-coherent network linking the entire optical spectrum to themicrowave standard. This advance also forms the basis for realization ofcontrollable optical waveforms which is important in ultrafast science.

[0046] Simple extension to use of other harmonically generated combfrequencies are specifically included in this invention. For example, 3times the red spectral output can be heterodyned against 2 times theblue spectral output, yielding the frequency offset f_(o) as desired,even though the necessary relative bandwidth of the laser system is nowreduced to only a factor of 1.5. Similarly, optical harmonic factors of4 and 3 can provide the offset frequency f_(o), even when the combbandwidth limits have a ratio of only 1.3:1. As disclosed above, use ofa frequency shifter, such as an AOM, in either heterodyne arm willdisplace the beat frequency phase comparison frequency away from theawkward region near zero frequency, modulus the repetition rate.

[0047] The present invention may therefore comprise a method ofstabilizing the phase of a carrier wave signal with respect to anenvelope of the pulses emitted by a mode-locked pulsed laser comprising:obtaining an optical output from the pulsed laser that has a bandwidththat spans at least one octave; separating a first frequency output fromthe optical output having a first frequency; separating a secondfrequency output from the optical output, the second frequency outputhaving a second frequency that is twice the frequency of the firstfrequency; frequency doubling the first frequency output of the pulsedlaser to produce a frequency doubled first output; frequency shiftingthe second frequency output by a predetermined amount to produce asecond frequency shifted output; combining the frequency doubled firstoutput and the second frequency shifted output to obtain a beatfrequency signal; detecting the beat frequency signal; using the beatfrequency signal to phase coherently stabilize the phase of the carrierwave signal relative to the envelope of the pulsed laser.

[0048] The present invention may further comprise a method ofstabilizing the phase of a carrier wave signal with respect to anenvelope of the pulses emitted by a mode-locked pulsed laser comprising:obtaining an optical output from the pulsed laser that has a bandwidththat spans at least one octave; separating a first frequency output fromthe optical output having a first frequency; separating a secondfrequency output from the optical output, the second frequency outputhaving a second frequency that is twice the frequency of the firstfrequency; frequency doubling the first frequency output of the pulsedlaser to produce a frequency doubled first output; frequency shiftingthe frequency doubled first output by a predetermined amount to producea frequency doubled and shifted first output; combining the secondfrequency output and the frequency doubled and shifted first output toobtain a beat frequency signal; detecting the beat frequency signal;using the beat frequency signal to phase coherently stabilize the phaseof the carrier wave signal relative to the envelope of the pulsed laser.

[0049] The present invention may further comprise a method ofstabilizing the phase of a carrier wave signal with respect to anenvelope of the pulses emitted by a mode-locked pulsed laser comprising:obtaining an optical output from the pulsed laser that has a bandwidththat spans at least one octave; separating a first frequency output fromthe optical output having a first frequency; separating a secondfrequency output from the optical output, the second optical frequencyoutput having a second frequency that is twice the frequency of thefirst frequency; frequency doubling the first frequency output of thepulsed laser to produce a frequency doubled first output; frequencyshifting one of the frequency doubled first output and the secondfrequency output by a predetermined amount to produce a frequencyshifted output; combining one of the frequency doubled first output andthe second frequency output with the frequency shifted output to obtaina beat frequency signal; detecting the beat frequency signal; using thebeat frequency signal to stabilize the phase of the carrier wave signalrelative to the envelope of the pulsed laser.

[0050] The present invention may further comprise a method ofstabilizing the phase of a carrier wave signal with respect to anenvelope of the pulses emitted by a mode-locked pulsed laser comprising:obtaining an optical output from the pulsed laser that has a bandwidththat spans at least one octave; separating a first frequency output fromthe optical output having a first frequency; separating a secondfrequency output from the optical output, the second frequency outputhaving a second frequency that is twice the frequency of the firstfrequency; frequency shifting the first frequency output by apredetermined amount to produce a frequency shifted first output;frequency doubling the frequency shifted first output of the pulsedlaser to produce a frequency shifted and doubled first output; combiningthe second frequency output and the frequency shifted and doubled firstoutput to obtain a beat frequency signal; detecting the beat frequencysignal; using the beat frequency signal to phase coherently stabilizethe phase of the carrier wave signal relative to the envelope of thepulsed laser.

[0051] The present invention may further comprise a method ofstabilizing the phase of a carrier wave signal with respect to anenvelope of the pulses emitted by a mode-locked pulsed laser comprising:obtaining an optical output from the pulsed laser that has a bandwidththat spans less than one octave; separating a first frequency outputfrom the optical output having a first frequency; separating a secondfrequency output from the optical output having a second frequency;multiplying the first frequency output of the pulsed laser by an integervalue N that is at least equal to 2 to produce a frequency multipliedfirst output; multiplying the second frequency output of the pulsedlaser by N-1 to produce a frequency multiplied second output; frequencyshifting the frequency multiplied second output by a predeterminedamount to produce a frequency multiplied second frequency shiftedoutput; combining the frequency multiplied first output and thefrequency multiplied second frequency shifted output to obtain a beatfrequency signal; detecting the beat frequency signal; using the beatfrequency signal to phase coherently stabilize the phase of the carrierwave signal relative to the envelope of the pulsed laser.

[0052] The present invention may further comprise a mode-locked pulsedlaser system that stabilizes the phase of a carrier wave signal withrespect to an envelope of the pulses emitted by the mode-locked pulsedlaser system comprising: a mode-locked pulsed laser that generates anoptical output; a beam splitter that separates a first frequency signalfrom the optical output, having a first frequency, from a secondfrequency signal of the optical output, the second frequency signalhaving a second frequency; a first frequency multiplier aligned with thefirst frequency signal that multiplies said first frequency signal by aninteger value N that is at least equal to 2 to produce a frequencymultiplied first signal; a second frequency multiplier aligned with thesecond frequency signal that multiplies said second frequency signal byN-1 to produce a frequency multiplied second signal; a frequency shifteraligned with the frequency multiplied second frequency signal thatfrequency shifts the frequency multiplied second frequency signal by apredetermined amount to produce a frequency multiplied second frequencyshifted signal; a beam combiner that combines the frequency multipliedfirst signal and the frequency multiplied second frequency shiftedsignal to obtain a beat frequency signal; a detector aligned to detectthe beat frequency signal; a control signal generator that generatescontrol signals in response to the beat frequency signal; aservo-controller that modifies an optical cavity of the pulsed laser inresponse to the control signals to change relative velocity of theenvelope and the carrier wave signal in the optical cavity.

[0053] The advantages of the present invention are that a compact andrelatively inexpensive laser system can be provided that locks the phaseof the envelope of the pulses of a mode-locked pulsed laser to the phaseof the carrier wave. In this fashion, the system can be used formetrology such that an inexpensive and highly accurate optical clockscan be produced. In addition, by controlling the relative phase betweenthe envelope and the carrier wave, peak signals can be generated havinghigh constant energy values. In this fashion, the pulse laser systemwill provide clear advantages in the field of extreme non-linear optics.These advantages include above-threshold ionization and high harmonicgeneration/x-ray generation with intense femtosecond pulses. Asdisclosed below, above-threshold ionization can be used to determineabsolute phase using circularly polarized light. The measurement of thepulse-to-pulse phase can also be used to measure x-ray generationefficiency using the intense very short pulses of the device of thepresent invention.

BRIEF DESCRIPTION OF THE DRAWINGS

[0054]FIG. 1 is a map of a portion of the electromagnetic spectrumshowing frequencies of several atomic and molecular referencetransitions and the frequency ranges of various sources for generatingthose frequencies. The frequency comb generated by mode-lockedfemtosecond lasers spans the visible region and can be transferred tothe infrared region by different frequency generation.

[0055]FIG. 2A is a schematic illustration of an optical frequency combgenerator (OFCG) that utilizes an intracavity electro-optic modulator.

[0056]FIG. 2B is a graph of the optical frequency comb that is generatedby the device of FIG. 2A.

[0057]FIG. 3 is a schematic illustration of the manner in which an OFCGis used to measure the absolute frequency of an iodine stabilized inNd:YAG frequency near 532 nm.

[0058]FIG. 4 is a diagram of laser intensity versus time thatillustrates a pulse train that is generated by locking the phase ofsimultaneous locking modes.

[0059]FIG. 5 is a diagram of a typical Kerr-lens mode-locked Ti:sapphirelaser.

[0060]FIG. 6 is a schematic illustration showing the manner in which anon-linear Kerr-lens acts to focus high intensities that are transmittedthrough an aperture while low intensities experience losses.

[0061]FIG. 7A is a time domain description of the pulses emitted by amode-locked laser.

[0062]FIG. 7B illustrates the Gaussian distribution in the frequencydomain of the comb lines that are generated by the pulsed laser of thepresent invention.

[0063]FIG. 8a through 8 d are graphs of wavelength versus dispersionillustrating the dispersion created by various optical fibers.

[0064]FIG. 9 illustrates the input and output spectra of an optic fiberthat may be used in accordance with the present invention.

[0065]FIGS. 10A, 10B and 10C illustrate the correspondence betweenoptical frequencies and heterodyne beats in the RF spectrum.

[0066]FIG. 10A illustrates the optical spectrum of a mode-locked laserwith modespacing Δ plus a single frequency laser at frequency f₁.

[0067]FIG. 10B illustrates the signals detected by a fast photodiode.The signals illustrate an RF spectrum with equally spaced modes due tothe mode-locked laser.

[0068]FIG. 10C illustrates the output in a typical experiment of the RFspectrum showing both the repetition beats and the heterodyne beats.

[0069]FIGS. 11A, 11B and 11C schematically illustrate the manner inwhich modes of a cavity depend on the cavity length (FIGS. 11A and 11B),and the manner in which the swivel angle of the mirror affects thefrequency (FIG. 11C).

[0070]FIG. 12 is a schematic circuit diagram of a circuit that actuatesa length-swivel PZT based on two error signal inputs (V₁ and V₂).

[0071]FIGS. 13A and 13B are schematic illustrations of implementationsof the present invention.*

[0072]FIG. 14A is the experimental configuration illustrating afrequency measurement using a self referenced comb.

[0073]FIG. 14B illustrates the experimental results of the experimentillustrated in FIG. 14A.

[0074]FIGS. 15A and 15B illustrate the results of a time domain crosscorrelation.

[0075]FIG. 15A shows a typical cross correlation between pulse i andpulse i+2 emitted from a laser along with the resulting envelope. Therelative phase is extracted by measuring the difference between the peakof the envelope and the nearest fringe.

[0076]FIG. 15B is a plot of the relative phase versus the offsetfrequency divided by the repetition rate. The linear results show aslope of 4% with a small overall phase shift that is attributed to thecorrelator.

[0077]FIG. 16A illustrates phase dependence.

[0078]FIG. 16B illustrates dependence on the pulsewidth.

[0079]FIG. 17 is a schematic illustration of a cross correlator.

DETAILED DESCRIPTION OF THE INVENTION

[0080] The present invention is based on the use of mode-locked lasers.To understand the present invention, it is therefore important to have athorough understanding of the operation of mode-locked lasers.

[0081] Mode-locked Lasers. The OFCGs described above actually generate atrain of short pulses. This is simply due to interference among modeswith a fixed phase relationship and is depicted in FIG. 4. FIG. 4 alsoillustrates the pulse train generated by locking the phase ofsimultaneous oscillating modes. In the upper panel, the output intensity400 is illustrated for one mode. In other words, the average intensityof one mode is constant. The intensity of two modes is shown at 402. Theintensity of three modes is shown at 404. For a fixed phase relationshipfor 30 modes interference results in a pulse train 406. For randomphases, the interference pattern 408 is illustrated. The mode spacingillustrated in FIG. 4 is 1 GHz.

[0082] The first OFCG was built to generate short optical pulses ratherthan for optical frequency synthesis or metrology. [See T. Kobayashi, T.Sueta, Y. Cho, and Y. Matsuo, App. Phys. Lett. 21, 341-343 (1972)].Later work provided even shorter pulses from an OFCG. [See G M.Macfarlane, A. S. Bell, E. Riis, and A. I. Ferguson, Opt. Lett. 21,534-536 (1996)].

[0083] A laser that can sustain simultaneous oscillation on multiplelongitudinal modes can emit short pulses; it just requires a mechanismto lock the phases of all the modes, which occurs automatically in anOFCG due to the action of the EOM. Lasers that include such a mechanismare referred to as “mode-locked” (ML). While the term “mode-locking”comes from this frequency domain description, the actual processes thatcause mode-locking are typically described in the time domain.

[0084] The inclusion of gain [S. A. Diddams, L. S. Ma, J. Ye, and J. L.Hall, Opt. Lett. 24, 1747-1749 (1999); K. P. Ho and J. M. Kahm, IEEEPhotonics Technol. Lett 5, 721-725 (1993)] and dispersion compensation[L. R. Brothers and N. C. Wong, Opt. Lett. 22, 1015-1017 (1997)] inOFCGs brings them even closer to ML lasers. Indeed the use of ML lasersas optical comb generators has been developed in parallel with OFCG,starting with the realization that a regularly spaced train of pulsescould excite narrow resonances because of the correspondence with a combin the frequency domain. [See T. W. Hänsch, in Tunable Lasers andApplications, edited by A. Mooradain, T. Jaeger and P. Stokseth(Springer-Verlag, Berlin, 1976); E. V. Baklanov and V. P. chebotaev,Soviet Journal of Quantum Electronics 7, 1252-1255 (1977); R. Teets, J.Eckstein, and T. W. Hansch, Phys. Rev. Lett. 38, 760-764 (1977); M. M.Salour, Rev. Mod. Phys. 50, 667-681 (1978)]. Attention was quicklyfocused on ML lasers as the source of a train of short pulses. [See J.N. Eckstein, A. I. Ferguson, and T. W. Hänsch, Phys. Rev. Lett. 40,847-850 (1978).; A. I. Ferguson and R. A. Taylor, Proc. SPIE 369,366-373 (1983); S. R. Bramwell, D. M. Kane, and A. I. Ferguson, Opt.Commun. 56, 112-117 (1985); S. R. Bramwell, D. M. Kane, and A. I.Ferguson, J. Opt. Soc. Am. B 3, 208 (1986); S. R. Bramwell, A. I.Ferguson, and D. M. Kane, Opt Lett. 12, 666-668 (1987); D. J. Wineland,J. C. Bergquist, W. M. Itano, F. Diedrich, and C. S. Weimer, in TheHydrogen Atom, edited by G. F. Bassani, M. Inguscio and T. W. Hänsch(Springer-Verlag, Berlin, 1989), p. 123-133; H. R. Telle, G. Steinmeyer,A. E. Dunlop, J. Stenger, D. H. Sutter, and U. Keller, Appl. Phys. B 69,327 (1999); T. Udem, J. Reichert, R. Holzwarth, and T. W. Hänsch, Opt.Lett. 24, 881-883 (1999); T. Udem, J. Reichert, R. Holzwarth, and T. W.Hänsch, Phys. Rev. Lett. 82, 3568-3571 (1999); J. Reichert, R.Holzwarth, T. Udem, and T. W. Hänsch, Opt. Commun. 172, 59-68 (1999); J.von Zanthier, J. Abel, T. Becker, M. Fries, E. Peik, H. Walther, R.Holzwarth, J. Reichert, T. Udem, T. W. Hänsch, A. Y. Nevsky, M. N.Skvortsov, and S. N. Bagayev, Opt. Commun. 166, 57-63 (1999); S. A.Diddams, D. J. Jones, L. S. Ma, S. T. Cundiff, and J. L. Hall, Opt.Lett. 25, 186-188 (2000); D. J. Jones, S. A. Diddams, J. K. Ranka, A.Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, Science 288,635-639 (2000); D. J. Jones, S. A. Diddams, M. S. Taubman, S. T.Cundiff, L. S. Ma, and J. L. Hall, Opt. Lett. 25, 308-310 (2000); S. A.Diddams, D. J. Jones, J. Ye, T. Cundiff, J. L. Hall, J. K. Ranka, R. S.Windeler, R. Holzwarth, T. Udem, and T. W. Hansch, Phys. Rev. Lett. 84,5102-5105 (2000); J. Reichert, M. Niering, R. Holzwarth, M. Weitz, T.Udem, and T. W. Hänsch, Phys. Rev. Lett. 84, 3232-3235 (2000); T. H.Yoon, A. Marian, J. L. Hall, and J. Ye, Phys. Rev. A 63, 011402 (2000)].The recent explosion of measurements based on ML lasers has been largelydue to development of the Kerr-lens-mode-locked (KLM) Ti:sapphire laser[D. K. Negus, L. Spinelli, N. Goldblatt, and G. Feugnet, in AdvancedSolid-State Lasers (OSA, 1991), Vol. 10; D. E. Spence, P. N. Kean, andW. Sibbett, Opt. Lett. 16, 42-44 (1991); M. T. Asaki, C. P. Huang, DGarvey, J. P. Zhou, H. C. Kapteyn, and M. M. Murnane, Opt. Lett. 18,977-979 (1993)] and its ability to generate pulses sufficiently short sothat the spectral width approaches an optical octave. Many recentresults have obtained a spectral width exceeding an octave by spectralbroadening external to the laser cavity. [See D. J. Jones, S. A.Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T.Cundiff, Science 288, 635-639 (2000); S. A. Diddams, D. J. Jones, J. Ye,T. Cundiff, J. L. Hall, J. K. Ranka, R. S. Windeler, R. Holzwarth, T.Udem, and T. W. Hansch, Phys. Rev. Lett. 84, 5102-5105 (2000); J. K.Ranka, R. S. Windeler, and A. J. Stentz, Opt. Lett. 25, 25-27 (2000)].

[0085] ML lasers have succeeded in generating much larger bandwidth thanOFCGs, which is very attractive. In addition, they tend to be “selfadjusting” in the sense that they do not require the active matchingbetween cavity length and modulator frequency that an OFCG does.Although the spacing between the longitudinal modes is easily measured(it is just the repetition rate) and controlled, the absolute frequencypositions of the modes is a more troublesome issue and requires somemethod of active control and stabilization. The incredible advantage ofhaving spectral width in excess of an octave is that it allows theabsolute optical frequencies to be determined directly from a cesiumclock [H. R. Telle, G. Steinmeyer, A. E. Dunlop, J. Stenger, D. H.Sutter, and U. Keller, Appl. Phys. B 69, 327 (1999); D. J. Jones, S. A.Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T.Cundiff, Science 288, 635-639 (2000); S. A. Diddams, D. J. Jones, J. Ye,T. Cundiff, J. L. Hall, J. K. Ranka, R. S. Windeler, R. Holzwarth, T.Udem, and T. W. Hansch, Phys. Rev. Lett. 84, 5102-5105 (2000)] withoutthe need for intermediate local oscillators.

[0086] 1. Introduction to Mode-locked Lasers. ML lasers generate shortoptical pulses by establishing a fixed phase relationship between all ofthe lasing longitudinal modes (see FIG. 4). [See J.-C. Diels and W.Rudolph, Ultrashort laser pulse phenomena: fundamentals, techniques, andapplications on a femtosecond time scale (Academic Press, San Diego,1996)]. Mode-locking requires a mechanism that results in higher netgain (gain minus loss) for a train of short pulses compared to CWoperation. This can be done by an active element, such as anacousto-optic modulator, or passively due to saturable absorption (realor effective). Passive ML yields the shortest pulses because, up to alimit, the self-adjusting mechanism becomes more effective than activemode locking, which can no longer keep pace with the ultrashort timescale associated with shorter pulses. [See E. P. Ippen, Appl. Phys. B58, 159-170 (1994)]. Real saturable absorption occurs in a material witha finite number of absorbers, for example a dye or semiconductor. Realsaturable absorption usually has a finite response time associated withrelaxation of the excited state. This typically limits the shortestpulse widths that can be obtained. Effective saturable absorptiontypically utilizes the nonlinear index of refraction of some materialtogether with spatial effects or interference to produce higher net gainfor more intense pulses. The ultimate limit on minimum pulse duration ina ML laser is due to an interplay between the ML mechanism (saturableabsorption), group velocity dispersion (GVD), and net gain bandwidth.

[0087] Because of its excellent performance and relative simplicity, theKerr-lens mode-locked Ti:sapphire (KLM Ti:S) laser has become thedominant laser for generating ultrashort optical pulses. A diagram of atypical KLM Ti:S laser is shown in FIG. 5. The laser comprises an outputcoupler 500 which generates an output 504. The laser cavity is formedfrom the output coupler 500 and the adjustable mirror 508. The Ti:Scrystal 502 is pumped by green light from a pump laser 510 such aseither an Ar⁺-ion laser (all lines or 514 nm) or a diode-pumped solidstate (DPSS) laser emitting 532 nm. Ti:S absorbs 532 more efficiently,so 4-5 watts of pump light is typically used from a DPSS laser, while6-8 W of light from an Ar⁺-ion laser is usually required. The Ti:Scrystal 502 provides gain and serves as the nonlinear material formode-locking. The prisms 506 compensate the GVD in the gain crystal.[See R. L. Fork, O. E. Martinez, and J. P. Gordon, Opt. Lett. 9, 150-152(1984)]. Since the discovery of KLM [D. K. Negus, L. Spinelli, N.Goldblatt, and G. Feugnet, in Advanced Solid-State Lasers (OSA, 1991),Vol. 10; D. E. Spence, P. N. Kean, and W. Sibbett, Opt. Lett. 16, 42-44(1991)], the pulse width obtained directly from the ML laser has beenshortened by approximately an order of magnitude by first optimizing theintracavity dispersion [M. T. Asaki, C. P. Huang, D. Garvey, J. P. Zhou,H. C. Kapteyn, and M. M. Murnane, Opt. Lett. 18, 977-979 (1993)], andthen using dispersion compensating mirrors [U. Morgner, F. X. Kartner,S. H. Cho, Y. Chen, H. A. Haus, J. G. Fujimoto, E. P. Ippen, V. Scheuer,G. Angelow, and T. Tschudi, Opt. Lett. 24, 411-413 (1999); D. H. Sutter,G. Steinmeyer, L. Gallmann, N. Matuschek, F. Morier-Genoud, U. Keller,V. Scheuer, G. Angelow, and T. Tschudi, Opt. Lett. 24, 631-633 (1999)]yielding pulses that are less than 6 fs in duration, i.e. less than twooptical cycles. A brief review is provided as to how a KLM laser works.While there are other ML lasers and mode-locking techniques, those arenot addressed because of the ubiquity of KLM lasers at the present time.Pulses of similar duration were achieved earlier [R. L. Fork, C. H. B.Cruz, P. C. Becker, and C. V. Shank, Opt. Lett. 12, 483-485 (1987)];however, this relied on external amplification at a low repetition, withpulse broadening and compression, which does not preserve a suitablecomb structure for optical frequency synthesis.

[0088] The primary reason for using Ti:S is its enormous gain bandwidth,which is necessary for supporting ultrashort pulses by the Fourierrelationship. The gain band is typically quoted as extending from 700 to1000 nm, although lasing can be achieved well beyond 1000 nm. If thisentire bandwidth could be mode-locked as a hyperbolic secant of Gaussianpulse, the resulting pulse width would be 2.5-3 fs. While this muchbandwidth has been mode-locked, the spectrum is far from smooth, leadingto longer pulses.

[0089] The Ti:S crystal also provides the mode-locking mechanism inthese lasers. This is due to the nonlinear index of refraction (Kerreffect), which is manifested as an increase of the index of refractionas the optical intensity increases. Because the intracavity beam'stransverse intensity profile is Gaussian, a Gussian index profile iscreated in the Ti:S crystal. A Gaussian index profile is equivalent to alens, hence the beam slightly focuses, with the focusing increasing withincreasing optical intensity. Together with a correctly positionedeffective aperture, the nonlinear (Kerr) lens 604, 606 can act as asaturable absorber, i.e. high intensities 602 are focused and hencetransmit fully through the aperture 610 while low intensities 600experience losses through aperture 610, as shown in FIG. 6. Since shortpulses produce higher peak powers, they experience lower loss, makingmode-locked operation favorable. While some KLM lasers include anexplicit aperture, the small size of the gain region can act as one.This mode-locking mechanism has the advantage of being essentiallyinstantaneous; no real excitation is created that needs to relax. It hasthe disadvantages of not being self-starting and requiring a criticalmisalignment from optimum CW operation.

[0090] Spectral dispersion in the Ti:S crystal due to the variation ofthe index of refraction with wavelength will result in spreading of thepulse each time it traverses the crystal. At these wavelengths, sapphiredisplays “normal” dispersion, where longer wavelengths travel faster theshorter ones. To counter-act this, a prism sequence is used in which thefirst prism spatially disperses the pulse, causing the long wavelengthcomponents to travel through more glass in the second prism than do theshorter wavelength components. [See R. L. Fork, O. E. Martinez, and J.P. Gordon, Opt. Lett. 9, 150-152 (1984)]. The net effect is to generate“anomalous” dispersion to counteract the normal dispersion in the Ti:Scrystal. The spatial dispersion is undone by placing the prism pair atone end of the cavity so that the pulse retraces its path through theprisms. It is also possible to generate anomalous dispersion withdielectric mirrors [R. Szipocs, K. Ferencz, C. Spielmann, and F. Krausz,Opt. Lett. 19, 201-203 (1994)]; these are typically called “chirpedmirrors.” They have the advantage of allowing shorter cavity lengths butthe disadvantage of less adjustability. Also, at present, they are stillhard to obtain commercially.

[0091] 2. Frequency Spectrum of Mode-locked Lasers. To successfullyexploit ML lasers for the generation of frequency combs with knownabsolute frequencies, it is necessary to understand the spectrum emittedby a mode-locked laser, how it arises and how it can be controlled.While it is always possible to describe the operation of these lasers ineither the time or frequency domain, the details of connecting the twoare rarely presented. Unless sufficient care is taken, it is easy formisunderstandings to arise when attempting to convert understanding inone domain into the other.

[0092] a. Time Domain Description of Pulses Emitted by a Mode-lockedLaser. Mode-locked lasers generate a repetitive train of ultrashortoptical pulses by fixing the relative phases of all of the lasinglongitudinal modes. [See A. E. Siegman, Lasers, (University ScienceBooks, Mill Valley Calif., 1986), p. 1041-1128]. Current mode-lockingtechniques are effective over such a large bandwidth that the resultingpulses can have a duration of 6 femtoseconds or shorter, i.e.,approximately two optical cycles. [See M. T. Asaki, C.-P. Huang, D.Garvey, J. Zhou, H. C. Kapteyn, M. M. Murnane, Opt. Lett. 18, 977(1993); U. Morgner, F. X. Kärtner, S. H. Cho, Y. Chen, H. A. Haus, J. G.Fujimoto, E. P. Ippen, V. Scheuer, G. Angelow, T. Tschudi, Opt. Lett.24, 411 (1999); D. H. Sutter, G. Steinmeyer, L. Gallmann, N. Matuschek,F. Morier-Genoud, U. Keller, V. Scheuer, G. Angelow, T. Tschudi, Opt.Lett. 24, 631 (1999)]. With such ultrashort pulses, the relative phasebetween peak of the pulse envelope and the underlying electric-fieldcarrier wave becomes significant. In general, this phase is not constantfrom pulse-to-pulse because the group and phase velocities differ insidethe laser cavity, as shown in the time domain diagram 700 of FIG. 7A. Todate, techniques of phase control of fs pulses have employed time domainmethods. [See L. Xu, Ch. Spielmann, A. Poppe, T. Brabec, F. Krausz, T.W. Hänsch, Opt. Lett. 21, 2008 (1996)]. However, these techniques havenot utilized active feedback and rapid dephasing occurs due to pulseenergy fluctuations and other perturbations inside the cavity. Activecontrol of the relative carrier-envelope phase prepares a stablepulse-to-pulse phase relationship, as presented below and dramaticallyimpacts extreme nonlinear optics.

[0093] As shown in FIG. 7A, a mode-locked laser emits a pulse 702 everytime the pulse circulating inside the cavity impinges on the outputcoupler. This results in a train of pulses separated by timeτ=I_(e)/v_(g) where I_(e) is the length of the cavity and v_(g) is thenet group velocity. Due to dispersion in the cavity, the group and phasevelocities are not equal. This results in a phase shift of the “carrier”wave with respect to the peak of the envelope for each round trip. Theshift between successive pulses is designated as Δφ. It is given by${\Delta \quad \varphi} = {\left( {\frac{1}{v_{g}} - \frac{1}{v_{p}}} \right)\quad l_{c}\omega_{c}\quad {mod}{\quad \quad}2\pi}$

[0094] where v_(p) is the intracavity phase velocity and ω_(c) is thecarrier frequency. This pulse-to-pulse shift is shown in FIG. 7A. Theoverall carrier-envelope phase of a given pulse, which obviously changesfrom pulse-to-pulse if Δφ≠0, includes an offset which does not affectthe frequency spectrum. There is no concern with this offset phase,although it is a subject of current interest in the ultrafast community.

[0095] b. Frequency Domain Considerations

[0096] 1) Comb Spacing. The frequency spectrum of the pulse trainemitted by a ML laser consists of a comb of frequencies 700. The spacingof the comb lines shown in FIG. 7B is simply determined by therepetition rate of the laser. This is easily obtained by Fouriertransforming a series of δ-function-like pulses in time. The repetitionrate is in turn determined by the group velocity and the length of thecavity.

[0097] 2) Comb Position. If all of the pulses have the same phaserelative to the envelope, i.e. Δφ=0, then the spectrum is simply aseries of comb lines with frequencies that are integer multiples of therepetition rate. However this is not in general true, due to thedifference between the group and phase velocities inside the cavity. Tocalculate the effect of a pulse-to-pulse phase shift on the spectrum, wewrite the electric field, E(t), of a pulse train. At a fixed spatiallocation, let the field of a single pulse be E₁(t)=Ê(t)e^(i(ω) ^(_(c))^(t+φ) ^(₀) ⁾. Then the field for train of pulses is $\begin{matrix}{{{E(t)} = {\sum\limits_{n}{{\hat{E}\left( {t - {n\quad \tau}} \right)}^{\text{?}}}}}\quad} \\{{= {\sum\limits_{n}{{\hat{E}\left( {t - {n\quad \tau}} \right)}^{\text{?}}}}},}\end{matrix}$ ?indicates text missing or illegible when filed

[0098] where Ê(t) is the envelope, ω_(c) is the “carrier” frequency, φ₀is the overall phase offset and τ is the time between pulses (for pulsesemitted by a mode-locked laser τ=t_(g), where t_(g) is the grouproundtrip delay time of the laser cavity). Applying the Fouriertransform, $\begin{matrix}{{E(\omega)} = {\int{\sum\limits_{n}{{\hat{E}\left( {t - {n\quad \tau}} \right)}^{\text{?}}^{- \text{?}}{t}}}}} \\{{= {\sum\limits_{n}{^{\text{?}}{\int{{\hat{E}\left( {t - {n\quad \tau}} \right)}^{\text{?}}{t}}}}}},}\end{matrix}$ ?indicates text missing or illegible when filed

[0099] letting {tilde over (E)}(ω)=∫Ê(t)e^(−int) dt and recalling theidentity

∫ƒ(x−a)e^(−laα)dx=e^(−laα)∫ƒ(x)e^(−laα)dx we obtain $\begin{matrix}{{E(\omega)} = {\sum\limits_{n}{^{\text{?}}^{- \text{?}}{\overset{\sim}{E}\left( {\omega - \omega_{e}} \right)}}}} \\{= {^{\text{?}}{\sum\limits_{n}{^{\text{?}}{{\overset{\sim}{E}\left( {\omega - \omega_{e}} \right)}.}}}}}\end{matrix}$ ?indicates text missing or illegible when filed

[0100] The significant components in the spectrum are the ones for whichthe exponential in the sum add coherently because the phase shiftbetween pulse n and n+1 is a multiple of 2π. Equivalently Δφ−ωτ=2m π.This yields a comb spectrum with frequencies$\omega_{m} = {\frac{\Delta \quad \varphi}{\tau} - \frac{2m\quad \pi}{\tau}}$

[0101] or, converting from angular frequency, f_(m)=mf_(rep)+δ whereδ=Δφ·f_(rep)/2πand f_(rep)=1/τ is the repetition frequency. Hence, theposition of the comb is offset from integer multiples of the repetitionrate by a frequency δ, which is determined by the pulse-to-pulse phaseshift. This is shown schematically in FIG. 7B.

[0102] 3. Frequency Control of the Spectrum. For the comb generated by aML laser to be useful for synthesizing optical frequencies, control ofits spectrum, i.e., the absolute position and spacing of the comb lines,is necessary. In terms of the above description of the output pulsetrain, this means control of the repetition rate, f_(rep), and thepulse-to-pulse phase shift, Δφ. Once the pulses have been emitted by thelaser, f_(rep) cannot be controlled. Δφ can be controlled by shiftingthe frequency of the comb, for example with an acousto-optic modulator.[See R. J. Jones, J. C. Diels, J. Jasapara, and W. Rudolph, Opt. Commun.175, 409-418 (2000)]. However it is generally preferable to control bothf_(rep) and Δφ by making appropriate adjustments to the operatingparameters of the laser itself. Some experiments only require control ofthe repetition rate, f_(rep). This can easily be obtained by adjustingthe cavity length.

[0103] Many experiments are simplified by locking both f_(rep) and δ. Todo so, both the round trip group delay and the round trip phase delaymust be controlled. Adjusting the cavity length changes both. If werewrite f_(rep) and δ in terms of round trip delays we find${f_{rep} = \frac{1}{t_{g}}};{\delta = {\frac{\omega_{c}}{2\pi \quad t_{g}}\left( {t_{g} - t_{p}} \right)}}$

[0104] where t_(g)=l_(c)/v_(g) is the round trip group delay andt_(p)=l_(c)/v_(p) is the round trip phase delay. Both t_(g) and t_(p)depend on l_(c), therefore another parameter must be used to controlthem independently. Methods for doing this will be discussed later. Theequation for δ may seem unphysical because it depends on ω_(c), which isarbitrary, however there is an implicit dependence on ω_(c) that arisesdue to the dispersion in v_(p) (and hence t_(p)) that cancels theexplicit dependence. Note that v_(p) must have dispersion forv_(p)≠v_(g) and that v_(g) must be constant (dispersionless) for stablemode-locked operation.

[0105] 4. Spectral Broadening. It is generally desirable to have a combthat spans the greatest possible bandwidth. At the most basic level,this is simply because it allows the measurement of the largest possiblefrequency intervals. Ultimately, when the output spectrum from a combgenerator is sufficiently wide, it is possible to determine the absoluteoptical frequencies of the comb lines directly from a microwave clock,i.e. without relying on intermediate phase locked oscillators. Thesimplest of these techniques requires a spectrum that spans an octave,i.e. the high frequency components have twice the frequency of the lowfrequency components. In the transform limit and for a spectrum with asingle peak, this bandwidth would correspond to a single cycle pulse,which has yet to be achieved. Thus it has been necessary to rely onexternal broadening of the spectrum. Fortunately, the optical heterodynetechnique employed for detection of single comb components is verysensitive, therefore it is not necessary to have a 3 dB bandwidth of anoctave. Detection is typically feasible even when the power at theoctave points is 10 to 30 dB below the peak. As noted above, additionaltechniques allow our measurement control system to be used even when thecomb frequency ratios are smaller than 2:1.

[0106] Self-phase modulation (SPM) occurs in a medium with a nonlinearindex of refraction, i.e. a third order optical nonlinearity. Itgenerates new frequencies, thereby broadening the spectrum of a pulse.This process occurs in the gain crystal of a ML laser and can result inoutput spectra that exceed the gain bandwidth. In the frequency domainit can be viewed as four-wave-mixing between the comb lines. The amountof broadening increases with the peak power per unit cross-sectionalarea of the pulse in the nonlinear medium. Consequently, optical fiberis often used as a nonlinear medium because it confines the opticalpower in a small area and results in an interaction length that islonger than could be obtained in a simple focus. While the small crosssection can be maintained for very long distances, the high peak powerin fact is typically only maintained for a rather short distance becauseof group velocity dispersion in the fiber, which stretches the pulse intime and reduces the peak power. Nevertheless, it has been possible togenerate an octave of usable bandwidth with ordinary single spatial modeoptical fiber by starting with very short and intense pulses from a lowrepetition rate laser.[See A. Apolonski, A. Poppe, G. Tempea, C.Spielmann, T. Udem, R. Holzwarth, T. W. Hänsch, and F. Krausz, Phys.Rev. Lett. 85, 740-743 (2000)]. The low repetition rate increases theenergy per pulse despite limited average power, thereby increasing thebroadening. The fiber was only 3 mm long. The pulses were pre-chirpedbefore being launched into the fiber so that the dispersion in the fiberwould recompress them.

[0107] The recent development of microstructured fiber has made itpossible to easily achieve well in excess of an octave bandwidth usingthe output from an ordinary KLM Ti:S laser. [See J. K. Ranka, R. S.Windeler, and A. J. Stentz, Opt. Lett. 25, 25-27 (2000); J. K. Ranka, R.S. Windeler, and A. J. Stentz, Opt. Lett. 25, 796-798 (2000)].Microstructured fibers consist of a fused silica core surrounded by airholes. This design yields a waveguide with a very high contrast of theeffective index of refraction between the core and cladding. Theresultant waveguiding provides a long interaction length with a minimumbeam cross-section. In addition, the waveguiding permits designing ofthe zero-point of the group velocity dispersion to be within theoperating spectrum of a Ti:S laser (for ordinary fiber thegroup-velocity-dispersion zero can only occur for wavelengths longerthan 1.3 microns). FIGS. 8a-8 d show the dispersion curves for severaldifferent core diameters. [See J. K. Ranka, R. S. Windeler, and A. J.Stentz, Opt. Lett. 25, 796-798 (2000)]. This property means that thepulse does not disperse and the nonlinear interaction occurs over a longdistance (centimeters to meters, rather than millimeters in ordinaryfiber).

[0108] Typical input 902 and output 900 spectra are shown in FIG. 9. Theoutput spectrum 900 is very sensitive to the launched power andpolarization. It is also sensitive to the spectral position relative tothe zero-GVD point and the chirp of the incident pulse 902. Because thefiber displays anomalous dispersion, pre-compensation of the dispersionis not required if the laser spectrum is centered to the long wavelengthside of the zero-GVD point (i.e., in the anomalous dispersion region).Because the pulse 902 is tuned close to a zero-GVD point, the outputphase profile is dominated by third-order dispersion. [See L. Xu, M. W.Kimmel, P. O'Shea, R. Trebino, J. K. Ranka, R. S. Windeler, and A. J.Stentz, in XII International Conference on Ultrafast Phenomena, editedby S. M. T. Elsacsser, M. M. Murnane and N. F. Scherer (Springer-Verlag,Charleston, S.C., 2000)].

[0109] Nonlinear processes in the microstructure fiber also producebroadband noise in the radio-frequency spectrum of a photodiode thatdetects the pulse train 1002 as shown in FIGS. 10A and 10B. This hasdeleterious effects on the optical frequency measurements describedbelow because the noise can mask the heterodyne beats 1010 and 1012shown in FIG. 10C. The noise increases with increasing input pulseenergy and appears to display a threshold behavior. This makes itpreferable to use short input pulses as less broadening, and hence lessinput power is required. The exact origin of the noise is currentlyuncertain, it may be due to guided acoustic-wave Brillouin scattering.[See A. J. Poustie, Opt. Lett. 17, 574-576 (1992); A. J. Poustie, J.Opt. Soc. Am. B 10, 691-696 (1993)].

[0110] Optical Frequency Measurements Using Mode-locked Lasers. Opticalfrequency measurements are typically made to determine the absoluteoptical frequency of an atomic, molecular or ionic transition.Typically, a single frequency laser is locked to an isolated transition,which may be among a rich manifold of transitions, and then thefrequency of the single frequency laser is measured. Mode-locked lasersare typically employed in the measurement of the frequency of the singlefrequency laser. This is performed by heterodyning the single frequencylaser 1000 against nearby optical comb lines 1002 of the mode-lockedlaser as shown in FIG. 10A. The resulting heterodyne RF signal 1010 and1012 that is shown in FIG. 10C contains beats at frequenciesf_(b)=|f₁−nf_(rep)−δ| where f₁ is the frequency of the single frequencylaser 1000, f_(rep) 1004 is the repetition rate of the ML laser, n aninteger and δ 1006 is the offset frequency of the ML laser (see previoussection). This yields, in the rf output of a photodetector, a pair ofbeats within every RF frequency interval between m f_(rep) and (m+1)f_(rep). One member of the pair arises from comb lines with n f_(rep)>f₁and the other from lines with n f_(rep)<f₁. Both, the beat frequenciesand f_(rep), can be easily measured with standard RF equipment.Typically, f_(rep) itself is not measured, but rather one of itsharmonics (10^(th) to 100^(th) harmonic), to yield a more accuratemeasurement in a given measurement time as the measurement phaseuncertainty is divided by the harmonic number. To map from these beatfrequency measurements to the optical frequency f₁, we need to know nand δ. Since n is an integer, it can be estimated using previousknowledge about f₁ to within ±f_(rep)/4. Typically f_(rep) is greaterthan 80 MHz, putting this requirement easily in the range ofcommercially available wavemeters, which have an accuracy of ˜25 MHz.Thus, measurement of δ is the remaining problem. This can be done bycomparison of the comb to an intermediate optical frequency standard,which is discussed below, or directly from the microwave cesium standarddescribed herein. Several such techniques for direct links betweenmicrowave and optical frequencies are described in this patent.

[0111] 1. Locking Techniques. Although measurement of f_(rep) and δ arein principle sufficient to determine an absolute optical frequency, itis generally preferable to use the measurements in a feedback loop toactively stabilize or lock one or both of them to suitable values. Ifthis is not done, then they must be measured simultaneously with eachother and with f_(b) to obtain meaningful results. The physicalquantities in the laser that determine f_(rep) and δ are describedabove. The technical details of adjusting these physical quantities aredescribed below.

[0112] The cavity length is a key parameter for locking the combspectrum. It is useful to examine a specific example of the sensitivityto the length. Consider a laser with a repetition rate of 100 MHz and acentral wavelength of 790 nm (˜380 THz). The repetition rate correspondsto a cavity length of 1.5 m which is 3.8 million wavelengths long(roundtrip). A length decrease of λ/2 gives an optical frequency shiftof exactly one order, +100 MHz, for all of the optical frequencycomponents. The corresponding shift of the 100 MHz repetition rate is(100 MHz)/(380 THz)=+2.63×10⁻⁷ fractionally or +26.31 Hz. Since thecavity length is most sensitive to environmental perturbations in bothfast (vibrations) and slow (temperature) timescales, good control of itwarrants close attention. Furthermore, length variations have a muchlarger effect on the optical frequency of a given comb line than on therepetition rate.

[0113] a. Comb Spacing. The comb spacing is given byf_(rep)=1/t_(g)=v_(g)/l_(c), where v_(g) is the round trip groupvelocity and l_(c) is the cavity length. The simplest way of lockingf_(rep) is by adjusting l_(c). Mounting either end mirror in the lasercavity on a translating piezo-electric actuator as shown in FIG. 5easily does this. The actuator is typically driven by a phase-lockedloop that compares f_(rep) or one of its harmonics to an external clock.For an environmentally isolated laser, the short time jitter in f_(rep)is lower than most electronic oscillators, although f_(rep) drifts overlong times. Thus the locking circuit needs to be carefully designed fora sufficiently small bandwidth so that f_(rep) does not have fast noiseadded while its slow drift is being eliminated.

[0114] Locking the comb spacing alone is sufficient for measurementsthat are not sensitive to the comb position such as measurement of thefrequency difference between two lasers (f_(L1) and f_(L2)). Forexample, suppose both frequencies of f_(L1) and f_(L2) are either higheror lower than their respective beating comb line. The difference ofthese two beats is taken using a double-balanced mixer and the resultingdifference frequency is f_(s)=f_(L1)=n f_(rep)−δ−(f_(L2)−mf_(rep)−δ)=(f_(L1)−f_(L2))−(m−n)f_(rep). Where n f_(rep)+δ<f_(L1) and mf_(rep)+δ>f_(L2), a minus sign arises due to the absolute value indetermining the beat frequencies (see expression for f_(b) above). Iff_(L1) and f_(L2) are known with accuracy better than f_(rep)/2, then(m−n) can be determined and hence f_(L1)−f_(L2) from f_(s).

[0115] b. Comb Position. The frequency of a given comb line is given byf_(n)=n f_(rep)+δ, where n is a large integer. Hence, simply changingthe cavity length can control the frequency of comb lines. However, thisalso changes the comb spacing, which is undesirable if a measurementspans a large number of comb lines. Controlling δ instead of f_(rep)allows a rigid shift of the comb positions, i.e. the frequency of allthe lines can be changed without changing the spacing.

[0116] The comb position depends on the phase delay, t_(g), and thegroup delay, t_(p). Each delay in turn depends on the cavity length. Inaddition, the comb position depends on the “carrier” frequency, which isdetermined by the lasing spectrum. To obtain independent control of boththe comb spacing and position, an additional parameter, besides thecavity length, must be adjusted.

[0117] A small rotation about a vertical axis (swivel) of the end mirrorof the laser in the arm that contains the prisms produces a controllablegroup delay as disclosed in FIG. 5. This is because the differentspectral components are spread out spatially across the mirror. Thedispersion in the prisms results in a linear relationship between thespatial coordinate and wavelength. Hence, the mirror swivel provides alinear phase with frequency, which is equivalent to a group delay. [SeeK. F. Kwong, D. Yankelevich, K. C. Chu, J. P. Heritage, and A. Dienes,Opt. Lett. 18, 558-560 (1993)]. The group delay depends linearly onangle for small angles. If the pivot point for the mirror corresponds tothe carrier frequency, then the effective cavity length does not change.The angle by which the mirror is swiveled is very small, approximately10⁻⁴ rad. If we assume that swiveling the mirror only changes the groupdelay by an amount αθ, where θ is the angle of the mirror and α is aconstant that depends on the spatial dispersion on the mirror and hasunits of s/rad, then we rewrite${f_{rep} = \frac{1}{\frac{l_{c}}{v_{g}} + {\alpha \quad \theta}}};{\delta = {\frac{\omega_{c}}{{2\pi}\quad}{\left( {1 - \frac{\frac{l_{c}}{v_{p}}}{\frac{l_{c}}{v_{g}} + {\alpha \quad \theta}}} \right).}}}$

[0118] From these equations we can derive how the comb frequenciesdepend on the control parameters, I_(c) and θ. To do so we will need thetotal differentials $\begin{matrix}{{f_{rep}} = {{{\frac{\partial f_{rep}}{\partial\theta}{\theta}} + {\frac{\partial f_{rep}}{\partial l_{c}}{l_{c}}}} = {{{- \frac{\alpha}{\left( {\frac{l_{c}}{v_{g}} + {\alpha \quad \theta}} \right)^{2}}}{\theta}} - {\frac{\frac{1}{v_{g}}}{\left( {\frac{l_{c}}{v_{g}} + {\alpha \quad \theta}} \right)^{2}}{l_{c}}}}}} \\{\cong {{{- \alpha}\frac{v_{g}^{2}}{l_{c}^{2}}{\theta}} - {\frac{v_{g}}{l_{c}^{2}}{l_{c}}}}} \\{{\delta} = {{\frac{\omega_{c}}{2\quad \pi}\frac{l_{c}\alpha}{v_{p}}\frac{1}{\left( {\frac{l_{c}}{v_{g}} + {\alpha \quad \theta}} \right)^{2}}{\theta}} - {\frac{\omega_{c}}{2\quad \pi}\frac{\frac{\alpha \quad \theta}{v_{p}}}{\left( {\frac{l_{c}}{v_{g}} + {\alpha \quad \theta}} \right)^{2}}{l_{c}}}}} \\{\cong {\frac{\omega_{c}}{2\quad \pi}\frac{v_{g}^{2}\alpha}{v_{p}l_{c}}{\theta}}}\end{matrix}$

[0119] where the final expressions are in the approximation that${\alpha \quad \theta}{\frac{l_{c}}{v_{g}}.}$

[0120] From this, we see that δ is controlled solely by θ. Theconcomitant change in f_(rep) can be compensated by changes in thecavity length.

[0121] Physically, these relationships can be understood by consideringhow the optical frequencies of individual modes depend on the length andswivel angle. This is shown schematically in FIGS. 11A, 11B and 11C foran example of a 1 cm long cavity. From the upper part it is clear thatthe dominant effect of changing the cavity length is the position changeof each optical mode, the repetition rate (spacing between modes)changes much less, by a ratio of the repetition rate/optical frequency.Thus the change in the repetition rate is only apparent for largerchanges in the length as is apparent by comparing FIGS. 11A and 11B.This is because repetition rate is multiplied up by the mode number toreach the optical frequency. Swiveling the mirror does not change thefrequency of the mode at the pivot point as is apparent from mode 10005of FIG. 11, but causes the adjacent modes to move in oppositedirections. This can be understood in terms of a frequency dependentcavity length, in this example it increases for decreasing frequencies.

[0122] c. Comb Position and Spacing. It is often desirable tosimultaneously control/lock both the comb position and spacing, or anequivalent set of parameters, say the position of two comb lines. In anideal situation, orthogonal control of f_(rep) and δ may be desirable toallow the servo loops to operate independently. If this cannot beachieved one servo loop will have to correct for changes made by theother. This is not a problem if they have very different responsespeeds. If the responses are similar, interaction between loops can lendto problems, including oscillation. If necessary, orthogonalization canbe achieved by either mechanical design in some cases, or by electronicmeans in all cases.

[0123] Note that it was earlier assumed that the pivot point of thetilting mirror corresponds to the carrier frequency. This is overlyrestrictive, as moving the pivot point will give an additional parameterthat can be adjusted to help orthogonalize the parameters of interest.The treatment can be generalized to include an adjustable pivot point byallowing l_(c) to depend on θ. The resulting analysis shows that it isimpossible to orthogonalize δ and f_(rep) by simple choice of the pivotpoint. Further examination of FIG. 11C makes it clear why this must beso. The comb line at (or near) the pivot point does not change itsfrequency when the swivel angle changes. This is inconsistent with beingable to control δ while holding f_(rep) constant, which is equivalent toa rigid shift of all of the comb lines (none are constant, thereforethere can be no pivot point).

[0124] Although δ and f_(rep) cannot be orthogonalized by choosing apivot, the analysis does show how to do so by electrical means. Thelength of the cavity merely needs to be made proportional to the swivelangle. This can be implemented electronically and amounts to invertingthe linear matrix equation connecting dδ and d f_(rep) to dθ and dl_(c).An electronic remedy also addresses the practical issue thatexperimental error signals generated to control the comb often contain amixture of two degrees of freedom. For example, the two error signalscan correspond to the position of a single comb line (usually withrespect to a nearby single frequency laser) and the comb spacing. A moreinteresting situation is when the error signals correspond to theposition of two comb lines. Typically, this will be obtained by beatinga comb line on the low frequency side of the spectrum with a singlefrequency laser and a comb line on the high frequency side with thesecond harmonic of the single frequency laser. In both of these cases,the error signals contain a mixture of δ and f_(rep), which in turn aredetermined by a mixture of the control parameters l_(c) and θ.

[0125] As will be evident below, having a pair of error signals thatcorrespond to the positions of two comb lines is the most interesting.[See J. Ye, J. L. Hall, and S. A. Diddams, Opt. Lett. 25, 1675 (2000)].This is obtained by beating the two comb lines against two singlefrequency lasers with frequencies f_(L1) and f_(L2). The beatfrequencies are given by f_(bi)=f_(L1)−(n_(i)f_(rep)+δ) with l=1,2 (forclarity we have assumed that f_(L1) is above the nearest comb line withindex n_(i)). Taking the differentials of these equations and invertingthe result we obtain $\begin{matrix}{{f_{rep}} = \left. \frac{{f_{b1}} - {f_{b2}}}{n_{2} - n_{1}}\rightarrow{\frac{1}{n}\left( {{f_{b1}} - {f_{b2}}} \right)} \right.} \\{{\quad \delta} = \left. \frac{{\frac{n_{1}}{n_{2}}{f_{b2}}} - {f_{b1}}}{1 - \frac{n_{1}}{n_{2}}}\rightarrow{{f_{b2}} - {2{f_{b1}}}} \right.}\end{matrix}$

[0126] where the expressions after the arrows are for f_(b2)=2f_(b1),i.e. we are using the fundamental and second harmonic of a single laserwith n being the index of the laser comb mode just below the fundamentalfrequency of the external laser. These equations can be combined withthose connecting dδ and d f_(rep) to dθ and dl_(c) to obtain$\begin{matrix}{{\theta} = {\frac{2\pi \quad v_{p}l_{e}}{\omega_{c}v_{g}^{2}\alpha}\left( {{f_{b2}} - {2{f_{b1}}}} \right)}} \\{{l_{e}} = {{- \frac{l_{e}^{2}}{v_{g}n}}\left( {{\left( {1 - {2{An}}} \right){f_{b1}}} + {\left( {{An} - 1} \right){df}_{b2}}} \right)}} \\{A = {\frac{2\pi \quad v_{p}}{\omega_{c}l_{c}}.}}\end{matrix}$

[0127] These equations directly connect the observables with the controlparameters for this configuration (note that the product An is of order1). Equations of a similar form have been derived without determiningthe values of the coefficients. [See J. Ye, J. L. Hall, and S. A.Diddams, Opt. Lett. 25, 1675 (2000)].

[0128] It is desirable to design an electronic orthogonalization circuitthat is completely general. Although expressions indicate what thecoefficients connecting the observables and control parameters shouldbe, typically the values of all of the parameters that appear in thecoefficients are not known. Furthermore, there may be technical factorsthat cause unwanted mixing of the control parameters or error signalsthat must be compensated. Such a coupling arises, for example, becausethe two transducers have differing frequency response bandwidths.Finally, a general circuit can readily be adapted to experimentalconfigurations other than the one discussed in detail above.

[0129] The electronic implementation depends on the actuator mechanisms.We employed a piezoelectric transducer (PZT) tube where an applicationof a transverse (between the inside and outside of the tube) voltageresults in a change of the tube length. By utilizing a split outerelectrode, the PZT tube can be made to bend in proportion to the voltagedifference between the two outer electrodes. The common mode voltage, orthe voltage applied to the inner electrode causes the PZT to change itslength, which is designated as “piston” mode. Mounting the end mirror ofthe laser cavity on one end of the PZT tube, allows it to be bothtranslated and swiveled, which correspond to changing the cavity length(l_(c)) and the mirror angle (θ).

[0130] The circuit shown in FIG. 12 allows a mixture of two error inputsto be applied to both the piston and swivel modes 1200 of the mirror1202. The coefficients a and b are adjusted via the potentiometers wherethe neutral midpoints correspond to a=b=0 and more generally$a,{b \propto \frac{R_{up} - R_{low}}{R_{up} + R_{low}}}$

[0131] where R₁ represents the resistance above or below the feedpoint.The gains of the amplifiers, g₁ and g₂, provide additional degrees offreedom. The length and swivel angle are related to the input voltagesby $\begin{matrix}\begin{matrix}{l_{c} \propto {{g_{1}V_{1}} + {g_{2}V_{1}} - {g_{2}{aV}_{2}}}} \\{\theta \propto {2{g_{2}\left( {{b\quad V_{1}} + V_{2}} \right)}}}\end{matrix} & {{{- 1} < a},{b < 1.}}\end{matrix}$

[0132] Thus if a=b=0, then l_(c) only responds to V₁ and θ only respondsto V₂. By adjusting a, l_(c) can be made to also respond to V₂, witheither sign. Similarly for b and V₁.

[0133] d. Other Control Mechanisms. In addition to tilting the mirrorafter the prism sequences, the difference between the group and phasedelays can also be adjusted by changing the amount of glass in thecavity [L. Xu, C. Spielmann, A. Poppe, T. Brabec, F. Krausz, and T. W.Hänsch, Opt. Lett. 21, 2008-2010 (1996)] or adjusting the pump power.[See L. Xu, C. Spielmann, A. Poppe, T. Brabec, F. Krausz, and T. W.Hänsch, Opt. Lett. 21, 2008-2010 (1996); R. Holzwarth, T. Udem, T. W.Hänsch, J. C. Knight, W. J. Wadsworth, and P. S. J. Russell, Phys. Rev.Lett. 85, 2264-2267 (2000)].

[0134] The amount of glass can be changed by moving a prism orintroducing glass wedges. Changing the amount of glass changes thedifference between group delay and phase delay due to the dispersion inthe glass. It has the disadvantage of also changing the effective cavitylength. Furthermore, rapid response time in a servo loop cannot beachieved because of the limitation of mechanical action on therelatively high mass of the glass prism or wedge.

[0135] Changing the pump power changes the power of the intracavitypulse. This has empirically been shown to change the pulse-to-pulsephase. [See. L. Xu, C. Spielmann, A. Poppe, T. Brabec, F. Krausz, and T.W. Hänsch, Opt. Lett. 21, 2008-2010 (1996)]. Intuitively, it might beexpected that this would lead to a change in the nonlinear phase shiftexperienced by the pulses as they traverse the crystal. However, acareful derivation shows that the group velocity also depends on thepulse intensity in such a way as to mostly cancel the phase shift. [SeeH. A. Haus and E. P. Ippen, submitted for publication, (2001)]. Thus itis not too surprising that the correlation between power and phase shiftis the opposite of what is expected from this simple picture. [See L.Xu, C. Spielmann, A. Poppe, T. Brabec, F. Krausz, and T. W. Hänsch, Opt.Lett. 21, 2008-2010 (1996)]. The phase shift is attributed to theshifting of the spectrum that accompanies changing the power. This willyield a changing group delay due to group velocity dispersion in thecavity. Such an effect is similar to the control of θ. Nevertheless, itis possible to achieve very tight locking due to the much larger servobandwidth afforded by an optical modulator, as compared to physicallymoving an optical element. [See R. Holzwarth, T. Udem, T. W. Hänsch, J.C. Knight, W. J. Wadsworth, and P. S. J. Russell, Phys. Rev. Lett. 85,2264-2267 (2000)]. Although the servo loop is stabilizing the frequencyspectrum of the mode-locked laser, it also reduces amplitude noise,presumably because amplitude noise is converted to phase noise bynonlinear processes in the laser. More work is needed to fullyunderstand the mechanism and exploit it for control of the comb.

[0136] The offset frequency of the comb can also be controlledexternally to the laser cavity by using and acouto-optic modulator. Thishas been exploited to lock the comb to a reference cavity. [See R. J.Jones, J. C. Diels, J. Jasapara, and W. Rudolph, Opt. Commun. 175,409-418 (2000)].

[0137] 2. Direct Optical to Microwave Synthesis Using a Self ReferencedSynthesizer. Direct optical synthesis from a microwave clock is possibleusing only a single mode-locked laser without an auxiliary singlefrequency laser. This is done by directly frequency doubling the longwavelength portion (near frequency f) of the octave-spanning spectrumand comparing to the short wavelength side (near frequency 2f). Thus itrequires more power in the wings of the spectrum produced by thefemtosecond laser than if an auxiliary single-frequency cw laser isused. However, the fact that many comb-lines contribute to theheterodyne signal means that a strong beat signal can be obtained evenif the doubled light is weak.

[0138] A self referenced synthesizer was first demonstrated at JILA byJones et al. [See D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R.S. Windeler, J. L. Hall, and S. T. Cundiff, Science 288, 635-639(2000)]. A diagram of the experiment is shown in FIGS. 13A and 13B.

[0139]FIG. 13A is a schematic block diagram of a device for implementingthe present invention. As shown in FIG. 13A, a pulsed laser 1300 maycomprise a mode-locked pulsed laser that generates an output 1302 thatproduces pulses having a wide bandwidth and an envelope that is on theorder of femtoseconds wide. The output 1302 is applied to a bandwidthbroadener 1304 that further broadens the bandwidth of the mode-lockedpulses. The bandwidth broadener 1304 may be eliminated if the output ofthe pulsed laser 1300 has a bandwidth that is sufficiently wide to coverat least one octave. Detector 1306 also detects the pulses of the output1302 and generates an electrical output signal 1308 that isrepresentative of the frequency of the repetition of the pulse envelope.A synthesizer 1310 is combined with the detector output 1308 to producea control signal 1312 that is applied to the translating servo 1314 thattranslates at least one of the mirrors in the mode-locked pulse laser1300 to control the overall length of the optical cavity of the pulsedmode-lock laser 1300. The broadened optical output signal 1316 from thebandwidth broadener 1304 is applied to a beam splitter 1318 that splitsthe optical signal 1316 into a first frequency output signal 1320 havinga first frequency f₁ and a second frequency output signal 1322 having asecond frequency f₂ which is twice the frequency of the first frequencyoutput signal. The first frequency output signal 1320 is applied to afrequency doubler 1324 that doubles the frequency of the first outputsignal to produce a frequency doubled first output 1326. Frequencyshifter 1328 shifts the frequency of the second frequency output signalby a predetermined amount under the control of control signal 1330produced by control signal generator 1332 to produce a second frequencyshifted output signal 1334. Optical combiner 1336 combines the frequencydoubled first output signal 1326 and the second frequency shifted outputsignal 1334 to produce a beat frequency signal 1338. Detector 1340detects the optical beat frequency signal 1338 and produces anelectronic beat frequency signal 1342 that is applied to the controlsignal generator 1332. The control signal generator 1332 generatescontrol signals 1330 and 1344. As indicated above, control signal 1330causes the second frequency output signal 1322 to be shifted by apredetermined amount under the control of control signal 1330. Asdisclosed below, the amount that the second frequency output signal 1322is shifted an integer fraction of the repetition frequency 1308. This isdone so that the phase of the carrier signal and the envelope can beadjusted so that δ can be made to be equal to zero. Control signalgenerator 1332 also generates control signals 1344 as indicated below todrive the tilting servo 1346. The tilting servo 1346 functions to adjustthe speed of the envelope relative to the carrier signal in the lasercavity by modifying the spatially dispersed spectrum that is produced byprisms located within the laser cavity of the pulsed mode laser 1300.

[0140] The frequency of the electrical signal 1342, f_(ele), produced bythe detector 1340 results from the heterodyne difference between the twooptical signals 1326 and 1334. This produces a series of signals withfrequencies such asf_(ele)=f₁₃₃₄−f₁₃₂₆=m₁f_(rep)+δ+f_(shift)−2(m₂f_(rep)+δ)=δ+f_(shift),where the last step occurs because m₁=2 m₂ for frequencies separated byan octave, f_(shift) is the frequency of the shift imposed by thefrequency shifter 1328. The other frequencies arise from the oppositeordering of the optical frequencies and m₁ and 2 m₂ differing by a smallinteger (i.e. not an exact octave). Thus the electrical signal containsthe frequency δ, which is needed to determine the absolute opticalfrequency of all of the comb lines and the pulse-to-pulsecarrier-envelope phase shift.

[0141] The octave-spanning spectrum can be obtained by externalbroadening in microstructure fiber as generally described below andshown in FIG. 13B. The output is spectrally separated using a dichroicmirror. The long wavelength portion is frequency doubled using aβ-barium-borate crystal. Phase matching selects a portion of thespectrum near 1100 nm for doubling. The short wavelength portion of thespectrum is passed through an acoustic-optic modulator (AOM). The AOMshifts the frequencies of all of the comb lines. This allows the offsetfrequency, δ, to be locked to zero, which would not otherwise bepossible due to degeneracy in the RF spectrum between the f-2fheterodyne beat signal and the repetition rate comb. The resultingheterodyne beat directly measures δ and is used in a servo loop to fixthe value of δ. The repetition rate is also locked to a frequencysynthesizer 1310 which in turn is referenced to an atomic clock. The endresult is that the absolute frequencies of all of the comb lines areknown with an accuracy limited only by the rf standard.

[0142] A more detailed discussion of the experimental implementation ofthe f-2f heterodyne system of the present invention with respect to FIG.13B is given below. The continuum output by the microstructure fiber1350 is spectrally separated into two arms by a dichroic beam splitter1352. The visible portion of the continuum 1358 (500-900 nm, containingf_(2n)) is directed through one arm that contains an acousto-opticmodulator 1356 (AOM). The near-infrared portion of the continuum 1360(900-1100 nm, containing f_(n)) traverses the other arm of theapparatus, passing through a 4 mm thick β-Barium-Boratefrequency-doubling crystal 1354. The crystal is angle-tuned toefficiently double at 1040 nm. The beams from the two arms are thenmode-matched and recombined by combiner 1362. The combined beam 1364 isfiltered with a 10-nm bandwidth interference filter 1366 centered at 520nm and focused onto an avalanche photo diode 1368 (APD). Approximately 5μW are incident on the APD 1368 from the arm 1370 containing the AOM,while the frequency doubling arm 1372 provides about 1 μW. The resultingRF beats 1374 are equal to ±(δ−f_(AOM)) where f_(AOM) is the drivefrequency of the AOM and is generated to be 7/8f_(rep). The RF beats1374 are then fed into a tracking oscillator 1376 that phase locks avoltage-controlled oscillator to the beat to enhance the signal to noiseratio by significantly reducing the noise bandwidth. From the trackingoscillator output 1378, an error signal is generated that isprogrammable to be ${\frac{m}{16}f_{rep}},$

[0143] thus allowing the relative carrier-envelope phase to be lockedfrom 0 to 2 π in 16 steps of π/8 for m=0,1, . . . 15.

[0144] The resulting comb was then used to measure the frequency of a778 nm single frequency ti:sapphire laser 1400 that was locked to the5S_(1/2) (F=3) 5D_(5/2) (F=5) two-photon transition in ⁸⁵Rb. This wasdone by combining the comb of a self referenced fiber continuum 1402with the single frequency laser using a 50-50 beam splitter 1404 asshown in FIG. 14A. A small portion of the spectrum near 778 nm isselected and the beat between the comb and the single frequency lasermeasured. A histogram of the measured frequencies is shown in FIG. 14B.Averaging over several days yielded a value of −4.2±1.3 kHZ from theCIPM recommended value.

[0145] As indicated above, the present invention uses a titanium-dopedsapphire (Ti:S) laser (shown in FIG. 13B) that is pumped with a singlefrequency, frequency-doubled Nd:YVO₄ laser operating at 532 nm. The Ti:Slaser generates a 90 MHz pulse train with pulse widths as short as 10 fsusing Kerr lens mode-locking (KLM). [See M. T. Asaki et al., Opt. Lett.18, 977 (1993)]. The output pulse spectrum is typically centered at 830nm with a width of 70 nm. To generate a 10 fs pulse, the normaldispersion of the Ti:S crystal is compensated by incorporating a pair offused silica prisms inside the cavity. [See R. L. Fork, O. E. Martinez,J. P. Gordon, Opt. Lett. 5, 150 (1984)]. It is important to note thatafter the second prism, the optical frequencies of the pulse arespatially resolved across the high-reflector mirror; this property willbe utilized to stabilize the absolute frequency of the laser.

[0146] As also indicated above, the relative carrier-envelope phase (Δφ)in successive pulses generated by mode-locked lasers is not constant dueto a difference between the group and phase velocities inside thecavity. As shown in FIG. 7, this is represented by the frequency offset,δ, of the frequency comb from f_(n=0)=0. Denoting the pulse repetitionrate as f_(rep), the relative phase is related to the offset frequencyvia 2πδ=Δφf_(rep). Thus, by stabilizing both f_(rep) and δ, Δφ can becontrolled. Toward this end, as shown in FIG. 13B, the high-reflectormirror 1358 (behind the prism) is mounted on a piezo-electric transducer(PZT) tube that allows both tilt and translation. By comparing a highharmonic of the pulse repetition rate with the output of a highstability RF synthesizer, a feedback loop can lock the repetition rate,f_(rep), by translating the mirror. Because the pulse spectrum isspatially dispersed across the mirror, tilting of this mirror provides alinear phase change with frequency (i.e., a group delay for the pulse),thereby controlling both the repetition rate and the offset frequency.[See J. Reichert, R. Holzwarth, Th. Udem. T. W. Hänsch, Opt. Commun.172, 59 (1999)]. The maximum required tilt angle is 10^(″4) rad,substantially less than the beam divergence, so cavity misalignment isnegligible.

[0147] To stabilize the offset frequency of a single mode-locked laser,without external information, it is useful to generate a full opticaloctave, although other methods and structures can be used that do notrequire a full octave, as explained herein. The typical spectral outputgenerated by the Ti:S laser used in these experiments spans 70 nm or 30THz, while the center frequency is approximately 350 THz, i.e., thespectrum spans much less than a full octave. Propagation through opticalfiber is commonly used to broaden the spectrum of mode-locked lasers viathe nonlinear process of self-phase modulation (SPM), based on theintrinsic intensity dependence of the refractive index (the Kerreffect). Optical fiber offers a small mode size and a relatively longinteraction length, both of which enhance the width of the generatedspectrum. However, chromatic dispersion in the optical fiber rapidlystretches the pulse duration, thereby lowering the peak power andlimiting the amount of generated spectra. While zero dispersion opticalfiber at 1300 nm and 1550 nm has existed for years, optical fiber thatsupports a single spatial mode and with zero dispersion near 800 nm hasbeen available only in the last year. In this work we employ a recentlydeveloped air-silica microstructure fiber that has zero group velocitydispersion at 780 nm. [See J. Ranka, R. Windeler, A. Stentz, Opt. Lett.25, 25 (2000)]. The sustained high intensity (hundreds of GW/cm²) in thefiber generates a stable, phase coherent continuum that stretches from510 to 1125 nm (at −20 dB) as shown in FIG. 9. Through four-wave mixingprocesses, the original spectral comb in the mode-locked pulse istransferred to the generated continuum. As described above, the offsetfrequency, δ, is obtained by taking the difference between 2f_(n) andf_(2n). FIG. 13B details this process of frequency doubling f_(n) in anonlinear crystal and combining the doubled signal with f_(2n) on aphoto-detector. The resulting RF heterodyne beat is equal to δ. Inactuality, the beat arises from a large family of comb lines, whichgreatly enhances the signal-to-noise ratio. After suitable processing(described below) this beat is used to actively tilt the high-reflectormirror, allowing us to stabilize δ to a rational fraction of the pulserepetition rate.

[0148]FIG. 13C is similar to FIG. 13A but further illustrates othermethods that can be used to decrease the amount of optical bandwidththat is required from the mode-locked pulse laser 1300 in accordancewith the present invention. As shown in FIG. 13C, the bandwidthbroadener 1304 can be eliminated if sufficient bandwidth is provided bythe mode-locked laser 1300. Only a portion of a full octave will berequired of the optical signal 1316. As shown in FIG. 13C, the opticalsignal 1316 is divided by beam splitter 1318 into a first signal (f₁)1320 and a second signal (f₂) 1322. The first signal (f₁) 1320 isapplied to a frequency multiplier 1321. The frequency multiplier 1321multiplies the first signal (f₁) 1320 by an integer value N not lessthan 2. The second frequency signal (f₂) 1322 is applied to a frequencymultiplier 1319. The second frequency signal (f₂) 1322 is multiplied byan integer value N−1. The output 1323 of the frequency multiplier 1319is (N−1)f₂. This output 1323 is then applied to the frequency shifter1328. The process then proceeds in accordance with the descriptionprovided with respect to FIG. 13A. The output of the frequencymultiplier 1321 is an output 1326 which is Nf₁. Well known electrooptical materials can be used for the frequency multipliers 1319, 1321.An advantage of the device illustrated in FIG. 13C is that the deviceonly requires a fraction of a full octave for optical signal 1316. Thisis because the ratio of the high frequency signal (f₂) 1322 to the lowfrequency signal (f₁) 1320 is (N−1)/N, which is less than a full octave.

[0149] As above, the frequencies of the electrical signal can becalculated. They is now given byf_(ele)=f₁₃₃₄−f₁₃₂₆=(N−1)(m₁f_(rep)+δ)+f_(shift)−N(m₂f_(rep)+δ)=δ+f_(shift),where the last step occurs because frequency f₂ 1322 is chosen such that${f_{2} \cong {\frac{N - 1}{N}f_{1}}},$

[0150] which means that $m_{2} = {\frac{N - 1}{N}{m_{1}.}}$

[0151] Again the electrical signal contains the frequency δ, which isneeded to determine the absolute optical frequency of all of the comblines and the pulse-to-pulse carrier-envelope phase shift.

[0152] Absolute Optical Frequency Metrology. In addition to applicationsin the time domain, the stabilized mode-locked laser system shown inFIGS. 13A and 13B has an immediate and revolutionary impact also inoptical frequency metrology as explained above with respect to FIGS. 14Aand 14B. As shown schematically in FIG. 7B, when both the f_(rep) (combspacing) and the offset frequency δ (comb position) are stabilized,lying underneath the broadband continuum envelope is a frequency combwith precisely defined intervals and known absolute frequencies. Bystabilizing f_(rep) in terms of the primary 9.193 GHz cesium standard,we can then use this frequency comb as a self-referenced “frequencyruler” to measure any optical frequency that falls within the bandwidthof the comb. With this technique, a direct link between the microwaveand optical domains is now possible using a single stabilized fs laser.

[0153] To demonstrate this application, results are presented using thisprocedure to measure a continuous wave (CW) Ti:S laser operating at 778nm and locked to the 5S_(1/2) (F=3) 5D_(5/2)(F=5) two-photon transitionin ⁸⁵Rb. The experimental setup is shown in FIG. 14A. A portion of thestabilized frequency comb is combined with the 778-nm stabilized Ti:Slaser and spectrally resolved using a 1200 lines/mm grating. Theheterodyne beat between the frequency comb and the CW stabilized Ti:Slaser is measured using a PIN diode positioned behind a slit that passes˜1 nm of bandwidth about 778 nm. By counting both the offset frequency,δ, and the heterodyne beat signal between the CW Ti:S and the comb(f_(beat)), the unknown frequency is determined byf_(unknown)=≠δ+nf_(rep)±f_(beat). The sign ambiguity of f_(beat) arisesbecause it is not known á priori whether the individual frequency combmember closest to the 778-nm laser is at a higher or lower frequency. Asimilar ambiguity exists for δ. As the 778-nm frequency is already knownwithin much better than f_(rep)/2=45 MHz, f_(unknown) is found by simplyincrementing or decrementing n and using the appropriate sign off_(beat) and δ. FIG. 14B displays one set of measurement resultsrelative to the CIPM (1997) recommended value of 385,285,142,378±5.0 kHzfor the Rb transition. [See T. Quinn, Metrologia, 36, 211 (1999)]. Thefirst demonstration of this technique was within the uncertainty of theCIPM (1997) value. The width of the Gaussian distribution leads one tosuspect the measurement scatter is most likely due to phase noise in theRb atomic clock used to stabilize the repetition rate of the laser. Onlyminimal environmental stabilization of the laser cavity was performed.With a higher quality reference clock, improved environmental isolationof the mode-locked laser cavity and higher bandwidth servo loops, loweramounts of scatter are expected, not only for data such as thatpresented in FIG. 14B, but also for time domain data presented belowwith respect to FIG. 15B.

[0154] The average of the frequency measurements over several days,giving −4.2±1.3 kHz from the CIPM (1997) value, agrees quite well with aprevious measurement of the JILA rubidium 2-photon stabilized referencelaser, in which an offset was measured of −3.2±3.0 kHz. In this previouswork, the position of the broadened fs comb was not locked but ratherthe comb position was calibrated with the fundamental and secondharmonic of a secondary, stabilized cw laser, which itself was measuredwith the octave-spanning comb.

[0155] These results demonstrate absolute optical frequency measurementswith a single mode-locked laser. This technique represents an enormoussimplification over conventional frequency metrology techniquesincluding multiplier chains, and even other fs methods, describedearlier. The tools described in this letter should make absolute opticalfrequency synthesis and measurement a common laboratory practice,instead of the heroic effort it has been heretofore.

[0156] Temporal Cross-Correlation. Verification of control of Δφ in thetime domain is obtained by interferometric cross correlation between twodifferent, not necessarily adjacent, pulses in the pulse train. [See L.Xu, Ch. Spielmann, A. Poppe, T. Brabec, F. Krausz, T. W. Hänsch, Opt.Lett. 21, 2008 (1996)]. In fact, we performed a time-averagedcross-correlation between pulses i and i+2 using the correlator shown inFIG. 17. A multi-pass cell 1700 in one arm of the correlator is used togenerate the required 20 ns delay. To minimize dispersion, the beamsplitter 1702 is a 2 μm thick polymer pellicle with a thin gold coating.To obtain a well-formed interferogram, the mirror curvatures and theirseparations were chosen to mode-match the output from both arms. Theentire correlator is in a vacuum chamber held below 300 m Torr tominimize the effect of the dispersion of air. The second ordercross-correlation is measured using a two-photon technique [J. K. Ranka,A. L. Gaeta, A. Baltuska, M. S. Pschenichnikov, D. A. Wiersma, Opt.Lett. 22, 1344 (1997)] by focusing the recombined beam with a sphericalmirror onto a windowless GaAsP photo-diode. The band gap of GaAsP islarge enough and the material purity high enough so that appreciablesingle photon absorption does not occur. This yields a pure quadraticintensity response with a very short effective temporal resolution.

[0157] A typical cross correlation is shown in FIG. 15A. To determineΔφ, the fringe peaks of the interferogram are fit to a correlationfunction assuming a Gaussian pulse envelope. From the fit parameters,the center of the envelope is determined and compared with the phase ofthe underlying fringes to find Δφ. A fit of the fringe peaks assuming ahyperbolic secant envelope produced nearly identical results. A plot ofthe experimentally determined relative phases for various offsetfrequencies, along with a linear fit of the averaged data, is given inFIG. 15B. These results show a small offset of 0.7±0.35 rad from thetheoretically expected relationship Δφ=4πδ/f_(rep) (the extra factor of2 results because the cross-correlator compares pulse i and i+2). Theexperimental slope is within 5% of the theoretically predicted value andclearly demonstrates our control of the relative carrier-envelope phase.Despite our extensive efforts to match the arms of the correlator, weattribute the phase offset between experiment and theory to a phaseimbalance in the correlator. The number of mirror bounces in each arm isthe same, and mirrors with the same coatings were used for 22 of the 23bounces in each arm. Nevertheless, because of availability issues, thereis a single bounce that is not matched. Furthermore, the large number ofbounces necessary to generate the delay means that a very small phasedifference per bounce can accumulate and become significant. Inaddition, the pellicle beam splitter will introduce a small phase errorbecause of the different reflection interface for the two arms.Together, these effects can easily account for the observed offset. Thegroup-phase dispersion due to the residual air only accounts for a phaseerror of ˜π/100. We believe this correlation approach represents thebest measurement strategy that can be made short of demonstration of aphysical process that is sensitive to the phase.

[0158] The uncertainty in the individual phase measurements shown inFIG. 15B apparently arises both from the cross-correlation measurementitself and from environmental perturbations of the laser cavity that arepresently beyond the bandwidth of our stabilizing servo loops. Indeed ameasurement in the frequency domain by counting a locked offsetfrequency δ=19 MHz with 1-second gate time revealed a standard deviationof 143 Hz, corresponding to a relative phase uncertainty of 13.7 μrad.The correlator uses a shorter effective gate time, which decreases theaveraging and hence increases the standard deviation. Nevertheless, theuncertainty in the time domain is 10³ to 10⁴ times larger than that inthe frequency domain (see below), indicating that the correlator itselfcontributes to the measurement uncertainty.

[0159] With pulses generated by mode-locked lasers now approaching thesingle cycle regime [U. Morgner et al., Opt. Lett. 24, 411 (1999); D. H.Sutter et al., Opt. Lett 24, 631 (1999)], the control of thecarrier-envelope relative phase that has been demonstrated in accordancewith the present invention is expected to dramatically impact the fieldof extreme nonlinear optics. This includes above-threshold ionizationand high harmonic generation/x-ray generation with intense femtosecondpulses. Above threshold ionization using circularly polarized light hasrecently been proposed as a technique for determining the absolutephase. [See D. Dietrich, F. Krausz, P. B. Corkum, Opt. Lett. 25, 16(2000)]. Measurements of x-ray generation efficiency also show effectsthat are attributed to the evolution of the pulse-to-pulse phase. [SeeC. G. Durfee et al., Phys. Rev. Lett. 83, 2187 (1999)].

[0160] High Precision Atomic and Molecular Spectroscopy and CoherentControl. A phase stable femtosecond comb represents a major step towardsultimate control of light fields as a general laboratory tool. Manydramatic possibilities are ahead. For high resolution laserspectroscopy, the precision frequency comb provides a tremendousopportunity for improved measurement accuracy. For example in molecularspectroscopy, different electronic, vibrational, and rotationaltransitions can be studied simultaneously with phase coherent light ofvarious wavelengths, leading to determination of molecular structure anddynamics with unprecedented precision. For sensitive absorptionspectroscopy, multiple absorption or dispersion features can be mappedout efficiently with coherent multi-wavelength light sources. Aphase-coherent wide-bandwidth optical comb can also induce the desiredmulti-path quantum interference effect for a resonantly enhancedtwo-photon transition rate. [See T. H. Yoon, A. Marian, J. L. Hall, andJ. Ye, Phys. Rev. A 63, 011402 (2000)]. This effect can be understoodequally well from the frequency domain analysis and the time domainRamsey-type interference. The multi-pulse interference in the timedomain gives an interesting variation and generalization of thetwo-pulse based temporal coherent control of the excited statewavepacket.

[0161] Stabilization of the relative phase between the pulse envelopeand the optical carrier should lead to more precise control of the pulseshape and timing, opening the door for many interesting experiments inthe areas of extreme nonlinear optics and quantum coherent control.Coherent control of atomic or molecular excited state populations isusually achieved by controlling the relative phase of pairs ormultiplets of pulses. If the phase of the pulse(s) that arrive later intime is the same as the excited state-ground state atomic phase, theexcited state population is increased, whereas if the pulse isanti-phased, the population is returned to the ground state. Withultrashort pulses, it is possible to achieve coherent control byinterference between pathways of different nonlinear order, for examplebetween 3 photon and 4 photon absorption. Such interference betweenpathways is sensitive to the absolute phase. This has been discussed formultiphoton ionization. [See E. Cormier and P. Lambropoulos, Euro. Phys.J. D 2, 15-20 (1998)]. For a pulse with a bandwidth that is close tospanning an octave, interference between 1 and 2 photon pathways will bepossible, thereby lessening the power requirements. A simulation of thisis shown in FIG. 16A, where the upper state population is plotted as afunction of absolute phase for the ideal 3-level system as shown in theinset. In FIG. 16B, the dependence on pulse width is shown. The effectis measurable for pulses that are far from spanning an octave. Howeverreal atoms do not have perfectly spaced levels, which will reduce theeffect, and this simple simulation ignores selection rules.

[0162] Optical Clocks. Measurement of absolute optical frequencies hassuddenly become a rather simple and straightforward task. Establishedstandards can now be easily re-calibrated [J. Ye, T. H. Yoon, J. L.Hall, A. A. Madej, J. E. Bernard, K. J. Siemsen, L. Marmet, J.-M.Chartier, and A. Chariter, Phys. Rev. Lett. 85, 3797 (2000)], andmeasurement precision has reached an unprecedented level. [See M.Niering, R. Holzwarth, J. Reichert, P. Pokasov, T. Udem, M. Weitz, T. W.Hansch, P. Lemonde, G. Santarelli, M. Abgrall, P. Laurent, C. Salomon,and A. Clairon, Phys. Rev. Lett. 84, 5496-5499 (2000)]. What is the nextstep? With the stability of the optical frequency comb currently limitedby the microwave reference used for phase locking f_(rep), directstabilization of comb components based on ultrastable optical referencesholds great promise. The initial demonstration of precision phasecontrol of the comb shows that a single cw laser (along with itsfrequency doubled companion output) can stabilize all comb lines(covering one octave of the optical frequency spectrum) to a level of 1Hz to 100 Hz at 1-s. [See J. Ye, J. L. Hall, and S. A. Diddams, Opt.Lett. 25, 1675 (2000)]. With control orthogonalization, we expect thesystem will be improved so that every comb line is phase locked to thecw reference below 1 Hz level. Now we can generate a stable microwavefrequency directly from a laser stabilized to an optical transition,essentially realizing an optical atomic clock. At the same time, anoptical frequency network spanning an entire optical octave (>300 THz)is established, with millions of frequency marks stable at the Hz levelrepeating every 100 MHz, forming basically an optical frequencysynthesizer. The future looks very bright, considering the superiorstability (10⁻¹⁵ at 1 s) offered by the optical oscillators based on asingle mercury ion and cold calcium atoms developed at NIST. [See K. R.Vogel, S. A. Diddams, C. W. Oates, E. A. Curtis, R. J. Rafac, J. C.Pergquist, R. W. Fox, W. D. Lee, and L. Hollberg, submitted forpublication, (2000)]. Indeed, within the next few years it will beamusing to witness friendly competitions between the Cs and Rb fountainclocks and various optical clocks based on Hg⁺, Ca or another suitablesystem.

[0163] Conclusion. The present invention has demonstrated stabilizationof the carrier phase with respect to the pulse envelope of ultrashortpulses produced by a mode-locked laser using a self-referencingtechnique that does not require any external optical input. The phasecan either be locked so every pulse has the identical phase, or made tovary so that every i^(th) pulse has the same phase. In the frequencydomain, this means that the broad spectral comb of optical lines haveknown frequencies, namely a simple (large) multiple of the pulserepetition frequency plus a user-defined offset. This is particularlyconvenient if the repetition rate of the laser is locked to an accuratemicrowave or RF clock because then the absolute optical frequencies ofthe entire comb of lines are known. These results will impact extremenonlinear optics [C. G. Durfee et al., Phys. Rev. Lett. 83, 2187 (1999);Ch. Spielmann et al., Science 278, 661 (1997)], which is expected todisplay exquisite sensitivity to electric field of the pulse.

[0164] The self-referencing technique also represents a dramatic advancein optical frequency metrology making measurement of absolute opticalfrequencies possible using a single laser. A mode-locked laser is usedwhich emits a stable train of pulses at repetition rate f_(rep).Corresponding to the temporal shortness of the pulse, there is acorresponding spectral bandwidth. If the laser spectrum is sufficientlybroad, either as directly emitted or utilizing an external broadeningdevice, such that the spectral extremes are separated by a factor of 2in frequency, the optical spectrum emitted by the laser can becompletely determined in terms of rf frequencies. This allows easycomparison to the cesium started, which has heretofore been extremelydifficult.

[0165] If the laser spectrum does not extend over a factor of two infrequency, but is still significantly broad, for example 28% or more, amodified self-referencing technique can be used. An optical harmonicgenerator, capable of generating the 4-th harmonic of its input, can beprovided with the red end of the spectrum of the laser beam. Anotheroptical harmonic generator, designed to generate another usefulharmonic, such as the 3 -rd harmonic, can be provided with the blue endof the spectrum of the laser beam. Either of these beams can befrequency-shifted, either before or after the harmonic generation. Thetwo harmonic beams, one having been shifted, are combined for opticalheterodyne detection using a suitable fast photodetector. The detectedbeat frequency contains the frequency offset of the mode-locked lasersystem, along with its aliases with the repetition frequency, as well asthe repetition frequency and its harmonics. Suitable electronics providephase-coherent locking of the offset frequency as a rational fraction ofthe repetition rate. The resultant laser field has two usefulproperties: 1) the carrier-envelope phase evolves in a deterministicmanner from one pulse to the next; 2) the optical spectrum is a comb ofharmonics of the repetition frequency as shifted by the (stabilized)offset frequency.

[0166] Other combinations of intrinsic optical bandwidth and harmonicmultiplications may be used: the principle is to multiply frequenciesfrom the spectral ends of the laser system's bandwidth by differentharmonic numbers so as to arrive at the same harmonic frequency (in theuv for the case of visible lasers) which enables heterodyne detection.The rf output is the laser offset frequency. According to the invention,a frequency shifter is used somewhere in this comparison chain todisplace the spectral region in which the heterodyne beat appears. Phaseknowledge of the repetition rate signal and the frequency offset allowsthe imposition of useful phase coherence on the laser.

[0167] The foregoing description of the invention has been presented forpurposes of illustration and description. It is not intended to beexhaustive or to limit the invention to the precise form disclosed, andother modifications and variations may be possible in light of the aboveteachings. The embodiment was therefore chosen and described in order tobest explain the principles of the invention and its practicalapplication to thereby enable others skilled in the art to best utilizethe invention in various embodiments and various modifications as aresuited to the particular use contemplated. It is intended that theappended claims be construed to include other alternative embodiments ofthe invention except insofar as limited by the prior art.

What is claimed is:
 1. A method of stabilizing the phase of a carrierwave signal with respect to an envelope of the pulses emitted by amode-locked pulsed laser comprising: obtaining an optical output fromsaid pulsed laser that has a bandwidth that spans at least one octave;separating a first frequency output from said optical output having afirst frequency; separating a second frequency output from said opticaloutput, said second frequency output having a second frequency that istwice the frequency of said first frequency; frequency doublingsaid-first frequency output of said pulsed laser to produce a frequencydoubled first output; frequency shifting said second frequency output bya predetermined amount to produce a second frequency shifted output;combining said frequency doubled first output and said second frequencyshifted output to obtain a beat frequency signal; detecting said beatfrequency signal; using said beat frequency signal to phase coherentlystabilize said phase of said carrier wave signal relative to saidenvelope of said pulsed laser.
 2. The method of claim 1 wherein saidstep of frequency shifting said second frequency output by apredetermined amount to produce a second frequency shifted outputfurther comprises applying an adjustable frequency input signal to anacousto-optic modulator that adjusts said second frequency shiftedoutput by a fractional portion of the repetition frequency of saidenvelope.
 3. The method of claim 1 wherein said step of frequencyshifting said second frequency output by a predetermined amount toproduce a second frequency shifted output further comprises applying anadjustable electric signal to an electro optic modulator.
 4. A method ofstabilizing the phase of a carrier wave signal with respect to anenvelope of the pulses emitted by a mode-locked pulsed laser comprising:obtaining an optical output from said pulsed laser that has a bandwidththat spans at least one octave; separating a first frequency output fromsaid optical output having a first frequency; separating a secondfrequency output from said optical output, said second frequency outputhaving a second frequency that is twice the frequency of said firstfrequency; frequency doubling said first frequency output of said pulsedlaser to produce a frequency doubled first output; frequency shiftingsaid frequency doubled first output by a predetermined amount to producea frequency doubled and shifted first output; combining said secondfrequency output and said frequency doubled and shifted first output toobtain a beat frequency signal; detecting said beat frequency signal;using said beat frequency signal to phase coherently stabilize saidphase of said carrier wave signal relative to said envelope of saidpulsed laser.
 5. The method of claim 4 wherein said step of frequencyshifting said frequency doubled first output by a predetermined amountto produce a frequency doubled and shifted first output furthercomprises applying an adjustable acoustic signal to an acousto-opticmodulator.
 6. The method of claim 4 wherein said step of frequencyshifting said frequency doubled first output by a predetermined amountto produce a frequency doubled and shifted first output furthercomprises applying an adjustable electric signal to an electro opticmodulator.
 7. A method of stabilizing the phase of a carrier wave signalwith respect to an envelope of the pulses emitted by a mode-lockedpulsed laser comprising: obtaining an optical output from said pulsedlaser that has a bandwidth that spans at least one octave; separating afirst frequency output from said optical output having a firstfrequency; separating a second frequency output from said opticaloutput, said second optical frequency output having a second frequencythat is twice the frequency of said first frequency; frequency doublingsaid first frequency output of said pulsed laser to produce a frequencydoubled first output; frequency shifting one of said frequency doubledfirst output and said second frequency, output by a predetermined amountto produce a frequency shifted output; combining one of said frequencydoubled first output and said second frequency output with saidfrequency shifted output to obtain a beat frequency signal; detectingsaid beat frequency signal; using said beat frequency signal tostabilize said phase of said carrier wave signal relative to saidenvelope of said pulsed laser.
 8. A method of stabilizing the phase of acarrier wave signal with respect to an envelope of the pulses emitted bya mode-locked pulsed laser comprising: obtaining an optical output fromsaid pulsed laser that has a bandwidth that spans at least one octave;separating a first frequency output from said optical output having afirst frequency; separating a second frequency output from said opticaloutput, said second frequency output having a second frequency that istwice the frequency of said first frequency; frequency shifting saidfirst frequency output by a predetermined amount to produce a frequencyshifted first output; frequency doubling said frequency shifted firstoutput of said pulsed laser to produce a frequency shifted and doubledfirst output; combining said second frequency output and said frequencyshifted and doubled first output to obtain a beat frequency signal;detecting said beat frequency signal; using said beat frequency signalto phase coherently stabilize said phase of said carrier wave signalrelative to said envelope of said pulsed laser.
 9. A mode-locked pulsedlaser system that stabilizes the phase of a carrier wave signal withrespect to an envelope of the pulses emitted by said mode-locked pulsedlaser system comprising: a mode-locked pulsed laser that generates anoptical output; a beam splitter that separates a first frequency signalfrom said optical output, having a first frequency, from a secondfrequency signal of said optical output, said second frequency signalhaving a second frequency that is twice the frequency of said firstfrequency; a frequency doubler aligned with said first frequency signalthat produces a frequency doubled first signal; a frequency shifteraligned with said second frequency signal that frequency shifts saidsecond frequency signal by a predetermined amount to produce a secondfrequency shifted signal; a beam combiner that combines said frequencydoubled first signal and said second frequency shifted signal to obtaina beat frequency signal; a detector aligned to detect said beatfrequency signal; a control signal generator that generates controlsignals in response to said beat frequency signal; a servo-controllerthat modifies the optical cavity of said pulsed laser in response tosaid control signals to change the relative velocity of said envelopeand said carrier wave signal in said optical cavity.
 10. The system ofclaim 9 further comprising: a non-linear self-phase modulator thatbroadens the bandwidth of said optical output of said pulsed laser to abandwidth that spans at least one octave.
 11. The system of claim 9 or10 wherein said non-linear self-phase modulator comprises an air-silicamicostructure optical fiber.
 12. The system of claim 9 or 10 whereinsaid frequency doubler comprises: a -barium-borate frequency doublingcrystal.
 13. The system of claim 9 wherein the mode-locked pulsed lasergenerates an optical output that has bandwidth that spans at least oneoctave.
 14. A method of stabilizing the phase of a carrier wave signalwith respect to an envelope of the pulses emitted by a mode-lockedpulsed laser comprising: obtaining an optical output from said pulsedlaser that has a bandwidth that spans less than one octave; separating afirst frequency output from said optical output having a firstfrequency; separating a second frequency output from said opticaloutput, having a second frequency; multiplying said first frequencyoutput of said pulsed laser by an integer value N that is at least equalto 2 to produce a frequency multiplied first output; multiplying saidsecond frequency output of said pulsed laser by N−1 to produce afrequency multiplied second output; frequency shifting said frequencymultiplied second output by a predetermined amount to produce afrequency multiplied second frequency shifted output; combining saidfrequency multiplied first output and said frequency multiplied secondfrequency shifted output to obtain a beat frequency signal; detectingsaid beat frequency signal; using said beat frequency signal to phasecoherently stabilize said phase of said carrier wave signal relative tosaid envelope of said pulsed laser.
 15. The method of claim 1, 4, or 14wherein said step of using said beat frequency signal to stabilize saidphase of said carrier wave signal relative to said envelope of saidpulsed laser further comprises: generating control signals in responseto said beat frequency to modify the optical cavity of said pulsed laserto change the velocity of said envelope and said carrier wave signal insaid optical cavity.
 16. The method of claim 15 wherein said step ofmodifying said optical cavity of said pulsed laser further comprises:inserting prisms in said optical cavity that spatially disperse thespectrum of said carrier wave signal; translating at least one of themirrors of said laser cavity in response to said control signals;tilting the mirror in said laser cavity that reflects said spatiallydispersed spectrum in response to said control signal.
 17. The method ofclaim 1, 4, or 14 wherein said predetermined amount is coherentlyderived from the repetition frequency of said pulsed laser.
 18. Themethod of claim 1, 4, or 14 wherein said step of obtaining an opticaloutput from said pulsed laser further comprises broadening said opticaloutput from said pulsed laser using an optical fiber located externallyfrom said optical cavity of said pulsed laser.
 19. The method of claim1, 4, or 14 wherein said step of obtaining an optical output from saidpulsed laser further comprises generating a broadened optical outputfrom said pulsed laser.
 20. The method of claim 14 wherein said step offrequency shifting said frequency multiplied second frequency output bya predetermined amount to produce a frequency multiplied secondfrequency shifted output further comprises applying an adjustablefrequency input signal to an acousto-optic modulator that adjusts saidfrequency multiplied second frequency shifted output by a fractionalportion of the repetition frequency of said envelope.
 21. The method ofclaim 14 wherein said step of frequency shifting said frequencymultiplied second frequency output by a predetermined amount to producea frequency multiplied second frequency shifted output further comprisesapplying an adjustable electric signal to an electro optic modulator.22. A mode-locked pulsed laser system that stabilizes the phase of acarrier wave signal with respect to an envelope of the pulses emitted bysaid mode-locked pulsed laser system comprising: a mode-locked pulsedlaser that generates an optical output; a beam splitter that separates afirst frequency signal from said optical output, having a firstfrequency, from a second frequency signal of said optical output, saidsecond frequency signal having a second frequency; a first frequencymultiplier aligned with said first frequency signal that multiplies saidfirst frequency signal by an integer value N that is at least equal to 2to produce a frequency multiplied first signal; a second frequencymultiplier aligned with said second frequency signal that multipliessaid second frequency signal by N−1 to produce a frequency multipliedsecond signal; a frequency shifter aligned with said frequencymultiplied second frequency signal that frequency shifts said frequencymultiplied second frequency signal by a predetermined amount to producea frequency multiplied second frequency shifted signal; a beam combinerthat combines said frequency multiplied first signal and said frequencymultiplied second frequency shifted signal to obtain a beat frequencysignal; a detector aligned to detect said beat frequency signal; acontrol signal generator that generates control signals in response tosaid beat frequency signal; a servo-controller that modifies an opticalcavity of said pulsed laser in response to said control signals tochange relative velocity between said envelope and said carrier wavesignal in said optical cavity.
 23. The system of claim 22 furthercomprising a non-linear self-phase modulator that broadens the bandwidthof said optical output to produce said predetermined bandwidth.
 24. Thesystem of claim 9, 10, or 22 wherein said beam splitter comprises: adichroic mirror.
 25. The system of claim 9, 10, or 22 wherein saidfrequency shifter comprises: an acousto-optic modulator.
 26. The systemof claim 9, 10, or 22 wherein said control device comprises:carrier-envelope phase locking electronics.
 27. The system of claim 9,10, or 22 wherein said detector comprises: an avalanche photodiode. 28.The system of claim 9, 10, or 22 wherein said servo-controllercomprises: a piezoelectric transducer tube that provides both tilt andtranslation.